First^Year  Mathematics 


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THE  UNIVERSITY  HIGH  SCHOOL 


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FIRST- YEAR  MATHEMATICS 
FOR    SECONDARY    SCHOOLS 


First- Year  Mathematics 

For  Secondary  Schools 


By 

GEORGE  WILLIAM  MYERS 

Professor  of  the  Teaching  of  Mathematics  and  Astronomy,   College  of 
Education  of  the  University  of  Chicago 

and 

WILLIAM  R.   WICKES  HARRIS   F.   MacNEISH 

ERNST  R.   BRESLICH  ERNEST  A.  WREIDT 

Instructors  in   Mathematics  in  the   University   High  School 
of   the     University    of    Chicago 


j-J  0    i?   Z 

School  of  Education  Manuals 
Secondary  Texts 


CHICAGO  , 

THE  UNIVERSITY  OF  CHICAGO  PRESS 
1907 


FtS  1908 


Copyright  1906  By 
Geoboe  W.  Myees 


First  edition  privately  printed  October,  1906 
Second  impression  published  April  1907 


Composed  and  Printed  By 

The  University  of  Chicagfo  Press 

Chicago,  Illinois.  U.  S.  A. 


TABLE  OF  CONTENTS 

PAGE 

Preface xi 

CHAPTER  I 

NUMBER  GENERALIZED 

§  I.  Uses  of  Positive  and  Negative  Number i 

CHAPTER  n 

THE   OPERATIONS   APPLIED  TO   POSITIVE  NUMBERS 

§  2.  Indicating  Arithmetical  Operations  Algebraically    ...        6 
CHAPTER  III 

THE   ARITHMETICAL   OPERATIONS   WITH   NUMBERS   REPRESENTED 
BY   LINES 

§  3.  Operations  with  Lines 11 

§  4.  Sums  and  Differences  of  Angles     ...  ....       13 

§  5.  Products  as  Rectangles 13 

CHAPTER  IV 

ADDITION,    SUBTRACTION,   AND   MULTIPLICATION  OF  POSITIVE 
AND    NEGATIVE    WHOLE    NUMBERS 

§  6.  Adding  Positive  and  Negative  Whole  Numbers      ...  15 

§  7.  Subtracting  Positive  and  Negative  Whole  Numbers      .     .  17 

§  8.  Multiplication  of  Positive  and  Negative  Whole  Numbers  18 

§  9.  Law  of  Signs  for  Multiplication      ........  23 

CHAPTER  V 

OPERATIONS   ON  FRACTIONAL  NUMBERS   GENERALIZED 

§10.  Unit-Fractions — Review  Problems         24 

§11.  Fractions  Having  the  Same  Nimieratora 25 

§12.  Addition  and  Subtraction  of  Fractions  in  General    ...  25 

§13.  Multiplication  and  Division  of  Fractions  in  General    .     .  26 

V 


vi  Contents 

CHAPTER  VI 

USES   OF  THE   EQUATION 

PAGE 

§14.  Problems 29 

§15.  The  Sum  of  the  Three  Angles  of  Any  Triangle  ....  31 

§16.  Angles  Made  by  Two  Intersecting  Straight  Lines    ...  32 

§17.  The  Simi  of  the  Angles  of  Polygons  32 

§18.  The  Exterior  Angles  of  Regular  Polygons 33 

CHAPTER  VII 

USES   OF  INEQUALITIES 

§19.  Laws  for  Use  of  Expressions  of  Inequality 35 

§20.  Problems  with  Inequalities     . 35 

CHAPTER  VIII 

THE   OPERATIONS  OF   ARITHMETIC   ABBREVIATION 

§21.  General   Arithmetic        37 

§22.  Laws  of  Percentage  and  Interest 39 

CHAPTER  IX 

THE   EVALUATION   OF   EXPRESSIONS 

§23.  The  Circle  and  Sphere 40 

§24.  Motion  and  Mensuration 41 

CHAPTER  X 

GEOMETRIC  REPRESENTATION   OF   QUANTITY 

§25.  Drawing  to  Scale 44 

§26.  Summary 49 

§27.  Exceptional  Cases 49 

§28.  Triangles  Having  the  Same  Shape  (Similar  Triangles)   .  50 

CHAPTER  XI 

EQUATION  APPLIED   TO   SIMPLE  PROBLEMS   ON  BEAMS 

§29.  Common  Uses  of  Forces     ....         55 

§30.  Experiments '.  56 

§31.  First  Law  of  Parallel  Forces 59 

§32.  Practical  Problems 59 


Contents  vii 

PAGE 

\^7,.  Turning-Tendencies  (Leverages) 6i 

§34.  Second  Law  of  Parallel  Forces 66 

§35.  Practical    Problems 66 

§36.  Solution  by  One  Unknown  Number 74 

§37.  Solution  by  Two  Unknown  Numbers 75 

CHAPTER  XII 

THE   SIMPLE   EQUATION 

§38.  The   Axioms 78 

§39.  Solution  of  Equations  by  Axioms 81 

CHAPTER  XIII 

THE   GRAPH 

§40.  Locating  Points  by  Means  of  Numbers     ....  83 

§41.  Picturing  or  Plotting  Laws  Connecting  Nimibers      .     .  85 

CHAPTER  XIV 

EQUATIONS   CONTAINING  FRACTIONS 

§42.  Freeing  an  Equation  of  Denominators 88 

§43.  Problems  in  Equations  Involving  Fractions 89 

§44.  Problems  Leading  to  Equations  Containing  Fractional 

and  Literal  Numbers 91 

CHAPTER  XV 

THE   FUNDAMENTAL  PROCESSES   APPLIED   TO  INTEGRAL 
ALGEBRAIC   EXPRESSIONS 

§45.  Addition  of  Integral  Algebraic  Expressions 97 

§46.  Subtraction  of  Integral  Algebraic  Expressions    ...  99 

§47.  Multiplication  of  Integral  Algebraic  Expressions  .     .     .  loi 

§48.  Geometrical  Representation  of  Special  Forms  of  Products  106 

§49;  Division  of  Integral  Algebraic  Expressions      ....  108 

§50.  Cases  in  which  w=»,  and  w<« no 

§51.  Exercises in 

§52.  Division  of  a  Polynomial  by  a  Monomial       in 

§53.  Division  of  a  Polynomial  by  a  Polynomial      .....  112 


viii  Contents 

CHAPTER  XVI 

FRACTIONS 

PAGE 

§54.  Fractions  Having  the  Same  Denominator 1 14 

§55.  Fractions  Having  DiflFerent  Denominators 115 

§56.  Multiplication  of  Fractions 117 

§57.  Division  of  Fractions 119 

CHAPTER  XVII 

EQUATIONS  INVOLVING  FRACTIONAL  FORMS 

§58.  A  Monomial  Factor       .     .  122 

§59.  A  Binomial  Factor              123 

§60.  Uses  of  Factoring  in  Solving  Equations 128 

CHAPTER  XVIII 

FRACTIONS   INVOLVING  FACTORABLE   FORMS 

§61.  Factoring  in  Reducing  Fractions  to  Lowest  Terms     .     .  131 

§62.  Factoring  in  Adding  and  Subtracting  Fractions     .     .     .  131 

§63.  Factoring  in  Multiplying  Fractions 132 

CHAPTER  XIX 

FACTORING 

§64.  Monomial  Factors 133 

§65.  The  Perfect  Square 136 

§66.  The  Difference  of  Two  Squares 138 

§67.  The  Trinomial  of  the  Form  x^  +  ax+b 140 

§68.  The  Trinomial  of  the  Form  crv^'  +  ftjc  +  c 142 

§69.  The  Sum  or  Difference  of  Two  Cubes 145 

§70.  The  Factor  Theorem 146 

CHAPTER  XX 

RATIO,   PROPORTION,   AND   SIMILARITY 

§71.  Examples  and  Definition       149 

§72.  Exercises  and  Problems 149 

§73.  Similar  Figures  and  Proportionality 153 

§74.  Problems  and  Applications          ...  155 


Contents  ix 

CHAPTER  XXI 

LINEAR  EQUATIONS  CONTAINING  TWO  UNKNOWN  NUMBERS 

PAGE 

§75.  Plotting  Linear  Equations 158 

§76.  Graphical  Solution  of  Simultaneous  Equations  of  the 

First  Degree 159 

§77.  Equivalent    Equations        159 

§78.  Inconsistent  Equations 160 

§79.  Elimination       160 

CHAPTER  XXII 

QUADRATIC  EQUATIONS 

§80.  Uses   of   Quadratics    and    Interpretation    of     Negative 

Roots       164 

§81.  Formal  Problems 165 

§82.  Problems  Leading  to  Quadratics 166 

CHAPTER  XXIII 

LOGARITHMS 

§83.  Logarithms  of  Exact  Powers  of  10 168 

§84.  Logarithms  of  Numbers  that  are  not  Exact  Powers  of  10  170 

§85.  Meaning  and  Use  of  a  Four-Place  Table  of  Logarithms  174 

§86.  To  Find  from  a  Table  the  Logarithm  of  a  given  Nimiber  174 
§87.  To  Find  from  a  Table  the  Number  that    Corresponds 

to  a  Given  Logarithm 175 

§88.  Questions  and  Problems 176 

§89.  Problems   for   Solution   by   the   Aid   of  Logarithmic 

Tables ...  179 


PREFACE 

!  In  an  institution  like  the  School  of  Education,  which  is 
founded  upon  the  principle  of  pedagogic  advance,  apology  for 
attempting  an  improvement  in  the  teaching  of  secondary 
mathematics  is  unnecessary.  A  few  words  explanatory  of  the 
nature  of  the  experiment  which  originated  the  following 
pages  will  however  not  be  out  of  place  here. 

A  very  little  study  of  the  plan  of  mathematical  instruction 
now  generally  in  vogue  in  secondary  schools  reveals,  among 
others,  these  serious  weaknesses: 

By  giving  a  whole  year  to  algebra  alone  before  beginning 
geometry,  as  is  the  custom,  young  pupils  are  required  to  take  up 
many  very  difficult  matters  in  algebra  before  doing  anything 
with  even  the  easier  and  more  vivid  concepts  of  elementary 
geometry.  The  damage  done  to  beginners  by  this  procedure 
would  exist  in  kind,  though,  perhaps,  in  less  degree,  if  the 
order  of  algebra  and  geometry  in  the  curriculum  were  reversed. 

Such  an  order  of  subjects  would  make  possible  the  grow- 
ing of  kindred  matters  of  geometry,  sometimes  inductively, 
sometimes  deductively,  and  more  or  less  informally,  out  of 
arithmetical  mensuration,  and  would  postpone  passing  to  the 
more  abstract  science  of  algebra  to  the  second  year.  A  unified 
body  of  mathematical  truth  carefully  graduated  as  to  intrinsic 
difficulties  and  touching  in  a  broader  and  more  vital  way  the 
interests  and  possibilities  of  youth  is  what  is  now  needed  and 
here  attempted. 

A  second  weakness  of  current  procedure  is  that,  without 
the  systematic  aid,  either  illustratively,  or  demonstratively,  of 
the  more  graphic  and  more  visual  presentation  of  the  similar, 
or  analogous,  mathematical  ideas  and  truths  of  geometry,  it 
requires  that  a  whole  year — and  that,  too,  the  most  immature 


xii  Preface 

year — ^be  given  to  that  particular  mathematical  subject  which 
in  its  nature  is  the  most  abstract  of  all,  and  is  most  difficult  of 
all  to  relate  to  the  life-interests  of  boys  and  girls.  This  weak- 
ness might  be  somewhat  mitigated — it  would  not  be  removed 
— by  a  reversal  of  the  order  of  algebra  and  geometry.  Cor- 
relation of  the  mathematical  subjects  with  one  another  will 
largely  eliminate  it. 

A  third  weakness  is  that  the  present  scheme  isolates,  both 
as  to  matter  and  method  of  treatment,  the  mathematical  sub- 
jects from  each  other,  and  from  the  things  which  to  youth  seem 
the  reahties  of  life,  for  so  long  a  time  during  the  early  stages 
of  the  high-school  career,  as  to  make  secondary  mathematical 
study  at  once  discouraging,  unscientific,  and  unpsychological. 
Geometry  being  much  more  readily  related  to  the  obvious 
things  of  the  world  and  of  life,  this  weakness  would  be 
partially  relieved  by  a  reversal  of  the  subjects  in  the  curriculum. 
The  only  plan  promising  adequate  relief  is  an  organic  per- 
meation of  the  mathematical  subjects  with  matters  of  real 
moment  to  modern  youth  and  to  modern  human  interests. 

These  are  only  a  few  of  the  ideas  that  actuated  the  mathe- 
matical faculty  to  take  a  step  in  organizing  a  body  of  mathe- 
matical matter  out  of  the  relevant  materials  of  arithmetic, 
algebra,  and  geometry.  This,  of  course,  is  only  a  correlation 
of  the  mathematical  subjects  among  themselves.  Not  to  de- 
part so  far  from  the  present  scheme  as  to  render  our  plan  of 
Httle  interest  to  other  teachers,  it  was  determined  to  organize 
our  material  around  an  algebraic  core.  Accordingly,  algebra 
very  largely  gives  trend,  unity,  and  character  to  the  work  of 
this  first  year. 

The  central  purpose  of  this  year's  work  is  to  lay  a  broad 
and  soUd  foundation  of  mathematical  concepts  and  elemental 
truths,  and  to  build  sohdly  upon  them  in  various  directions, 
completing  very  definitely  a  considerable  body  of  algebra,  and 
accomplishing  meanwhile,  on  the  side,  a  rounding  out,  from 


Preface  xiii 

a  higher  point  of  view,  of  elementary-school  mathematics,  and 
doing  most  of  that  time-consuming  preliminary  work  necessary 
to  induct  beginners  into  the  ideas,  method,  and  spirit  of  geom- 
etry. 

Some  of  the  algebraic  subjects  are  given  only  a  first  treat- 
ment, to  be  completed  later,  and  many  geometrical  matters 
are  given  a  sufficiently  full  treatment  for  secondary  schools. 
Unessential  details,  artificial  complexities,  and  logical  over- 
niceties  are  omitted,  to  make  room  for  what  is  essential  and 
comprehensible  to  beginners  in  mathematical  reasoning.  The 
thought-values  of  the  work  are  stressed  throughout,  the  neces- 
sary technique  being  made  subsidiary  and  auxiliary  to  think- 
ing. Not  rules,  but  reasons,  in  the  early  stages  of  the  work, 
and  later,  when  rules  become  necessary,  rules  with  reasons,  are 
the  guiding  precepts. 

Plan  of  work  on  manuscript. — In  the  preparation  of 
the  manuscript  these  four  ideas  are  agreed  upon  by  its 
authors: 

1.  Departure  from  the  well-matured  and  long-practiced 
procedure  of  successful  teachers  is  allowed  only  when  it  is  sub- 
stantiated by  clear  and  cogent  reasons,  and  can  be  seen  to 
promise  a  distinct  gain  to  the  learner.  Pure  a  priori  argument 
is  not  sufficient. 

2.  Competent  and  well-supported  opinion  from  every  ob- 
tainable source,  inside  or  outside  of  the  faculty,  as  to  the  scien- 
tific and  practical  merit  of  proposed  plans,  is  invited  and  will 
be  fully  considered  with  a  view  to  acting  upon  it. 

3.  The  body  of  the  subject-matter  worked  out  is  to  be  re- 
garded, not  as  a  finished  product,  but  merely  as  a  stage  of 
study  of  the  problem  of  unifying  mathematics  in  the  secondary 
curriculum. 

4.  Youths,  no  less  than  children,  look  upon  the  doing  of 
things  as  of  most  worth  to  them.  Accordingly,  the  principles 
of  mathematics  are  developed  largely  through  the  working  of 


xiv  Preface 

problems  and  the  generalizing  of  processes.  Inductive  methods 
are  widely — not  exclusively,  nor  mainly — used. 

The  plan,  of  which  this  book  is  a  part,  now  under  way  in 
the  University  High  School  seeks  to  a  very  great  extent  to 
deduce  the  more  important  mathematical  conclusions  directly 
from  the  axioms.  No  attempt  is  made  in  the  initial  stages  to 
disentangle  and  to  render  precise  and  consistent  a  necessary 
and  suflEicient  body  of  axioms  for  algebra  and  geometry.  Nor 
does  it  resort  to  the  other  alternative,  so  generally  resorted  to 
in  the  texts,  of  stating  a  necessary  and  sufficient  number  of 
axioms  and  calling  upon  the  pupil  to  take  them  on  faith.  Stu- 
dents are  at  the  outset  allowed  to  use  a  very  rich  body  of  axioms. 
The  body  of  fundamental  notions  with  which  the  work  pro- 
ceeds for  a  time,  is  sufficient,  but  not  necessary.  After  all  is 
said,  axioms  and  postulates  are  only  assumed  truths,  and  no 
harm  is  done  in  allowing  many  propositions,  which  seem 
obvious  to  the  uninitiated,  such  as  "All  straight  (or  right)  an- 
gles are  equal,"  or  "From  a  point  on  a  line  a  perpendicular 
can  be  drawn,"  to  be  taken  as  axioms — they  ought,  indeed,  to 
be  so  taken — until  pupils  get  far  enough  into  the  subject  both 
to  know  and  to  appreciate  what  the  talk  is  about  when  their 
proof  is  undertaken.  Later  these  same  propositions  will  be 
proved.  It  must  be  added  also  that  pupils  must  always  be 
made  clearly  aware  of  what  is  being  assumed  in  an  argument. 

From  what  has  been  said  it  appears  that  the  pupil  assists 
in  the  process  of  disengaging  and  condensing  the  elemental 
truths,  and  through  this  assistance  he  comes  to  a  sense  of  their 
fundamental  and  necessary  character.  When  a  step  in  an  ar- 
gument, or  a  demonstration,  is  forgotten,  instead  of  resorting 
to  the  broken  chain  of  an  author's  arbitrary  sequence,  or, 
worse  still,  to  a  set  proof,  Hke  a  true  initiate  in  the  science, 
the  pupil  recurs  to  the  axioms  and  undertakes  to  draw  the  for- 
gotten argument  from  this  source.  The  attitude  of  mind  of 
the  learner  is  at  all  times  the  attitude  of  the  research  student. 


Preface  xv 

and  not  that  of  one  being  merely  instructed.  This  not  infre- 
quently brings  the  learner  upon  an  unexpected  truth  and  gives 
him  a  little  of  the  foretaste  of  discovery.  Only  the  teacher  can 
appreciate  how  wholesome  an  influence  this  spirit  infuses  into 
the  class. 

In  conclusion,  it  may  be  well  to  state  to  any  persons  into 
whose  hands  it  may  chance  to  fall  that  this  little  volume  has 
been  used  in  mimeograph  form  during  the  past  year  in  the 
entering  classes  of  the  University  High  School  and  results  were 
good  enough  to  lead  the  mathematical  faculty  to  desire  to  use 
it  again  in  preference  to  the  standard  texts.  It  is  herewith 
printed,  not  out  of  any  desire  of  the  authors  to  pubhsh  a  book, 
but  simply  to  get  the  matter  into  better  form  for  use  in  their 
own  classes,  and  to  hold  what  it  contains  in  usable  form  while 
the  second-year  work  is  being  organized.  Criticism,  sugges- 
tion, or  co-operation  from  any  competent  source  will  be  gladly 
welcomed  by  the  authors  to  the  end  that  the  finished  form, 
when  it  does  appear,  may  be  as  largely  and  widely  serviceable 
as  possible. 

The  Authors 

Chicago,  August,  1906 


i^-^-  > 


CHAPTER  I 

NUMBER  GENERALIZED 

§1.  Uses  of  Positive  and  Negative  Number 

1.  The  top  of  the  mercury  column  of  a  thermometer  stands 
at  o°  at  the  beginning  of  an  hour.  The  next  hour  it  rises  5° 
and  the  next  3°.     What  does  the  thermometer  read  ? 

2.  If  the  mercury  stands  at  0°,  and  rises  8°,  then  falls  5°, 
what  does  the  thermometer  read  ? 

3.  Denoting  a  rise  of  10°,  or  of  x°  in  the  thermometer  by 
R  10°,  or  by  R  x°,  and  a  fall  of  10°,  or  of  x°,  by  F  10°,  or  F  x°, 
give  the  readings  of  the  thermometer  after  the  following  changes, 
if  the  top  of  the  column  reads  0°  at  the  start: 


^ 


V 


(i)  R    8°  f< 

d11ow( 

^dbyR    5°; 

(7)  R    6°  fc 

)llowe 

dbyF    5°; 

(2)  R  12° 

byF    9°; 

(8)  R    6° 

byF    6°; 

(3)  R  16° 

by  F  12°; 

(9)F    6° 

byR    7-; 

(4)  F  13° 

byF    f; 

(10)  R  13° 

by  F  18°; 

(S)F    8° 

byR    6°; 

(II)  F    5° 

byR    x°; 

(6)  F  17° 

by  R  10°; 

(12)  R    a° 

byF    b°. 

4.  If  the  change  in  the  mercury  column  is  a  rise,  a  positive 
or  plus  (+)  sign  will  be  written  before  the  number  that  de- 
notes the  amount  of  the  change.  If  the  change  is  a  jail,  a 
negative  or  minus  (— )  sign  will  be  written  before  the  number. 
If  the  reading  at  the  start  is  0°,  give  the  readings  after  these 


changes: 

(i)+iof 

olio  wed  by  +  2 ; 

(8)  —20  followed  by  +20; 

(2)  +10 

by  —  2 

(9)  +12        ' 

by  -12; 

(3)  +20 

by  -18 

(10)  +  9        ' 

by -12; 

(4)  -25 

by  -  8 

(11)  +  X 

by  +  y; 

(s)  -16 

by  +16 

(12)  +  a 

by  -  x; 

(6)  +  9 

by  —10 

(13)  +  a       ' 

by  -  a; 

(7)  -20 

by  +18 

(14)  -  a 

'        by  —  ic. 

2  First- Year  Mathematics 

5.  A  bicyclist  starts  from  a  point  and  rides  18  miles  due 
northward  (  +  18  mi.)  then  10  mi.  due  southward  (  —  10  mi.); 
how  far  is  he  then  from  the  starting  point  ? 

6.  State  how  far  and  in  what  direction  from  the  starting 
point  a  bicycUst  would  be  after  rides  indicated  by  each  of  these 
pairs  of  records : 

(i)  +10  mi.  then  —  8  mi.;  (3)  +100  mi.  then  +50  mi.; 

(2)  >^  20  mi.     "     +20  mi.;  (4)  +     ami.     "     +  6  mi. 

7.  How  far  and  in  what  direction  from  the  starting  point 
is  a  traveler  who  goes  eastward  (+)  or  westward  (— )  as  shown 
by  these  pairs  of  numbers: 


(i)  +16  mi.  then  —  6  mi. 

(2)  -18  mi.     "     +28  mi. 

(3)  ~^  ^-     "     +3  ^• 


(4)  +a  mi.  then  +c  mi.; 

(5)  4-wmi.     "     — »mi.; 

(6)  —  w  mi.     **     +nnii. 

8.  A  car  in  the  middle  of  a  moving  train  is  drawn  forward 
with  a  force  of  8  tons  and  at  the  same  time  it  is  pulled  back- 
ward with  a  force  of  7^  tons.  The  two  forces  together  are 
equal  to  what  single  force  ? 

9.  Denoting  a  forward  pulling  force  by  F  and  a  backward 
by  B,  give  amount  and  direction  of  a  single  force  equal  to 
each  of  these  pairs  of  forces: 

(1)  F  14  oz.  with  B    6  oz.;  (3)  F  25  tons  with  B  15  tons; 

(2)  F  20  lb.    "      B  12  lb.;  (4)  B  25  tons     ''     F  40  tons. 

10.  Denoting  forward-pulling  by  the  positive  or  plus  (+) 
sign  and  back-pulling  forces  by  the  negative  or  minus  (— ) 
sign,  give  the  single  force  which  is  equal  to  each  of  these  pairs 
of  forces: 


(i)  -f2o  and  —12 

(2)  -1-20    "     —20 

(3)  -15    "     -  8 


(4)  —15  and  -t-  8;        (7)  +a  and  +b; 

(5)  -12    "     -12;        (8)  +a    "     -b- 

(6)  -h  X     "     -12;        (9)  -a    "     -b. 


1 1 .  A  toy  balloon  pulls  upward  with  a  force  of  9  oz.     If  a 


Number  Generalized  3 

weight  of  6  oz.  is  attached,  will  the  balloon  rise  or  fall  ?    With 
what  force  ? 

12.  Call  upward  forces  positive,  or  plus  (+),  and  down- 
ward forces  negative,  or  minus  (— ).  State  what  single  force 
will  have  the  same  effect  as  these  pairs: 


(i)   +17  lb.  and  —  7  lb. 

(2)  +17  lb.    "     -10  lb. 

(3)  -231b.    -     -10  lb. 


(4)  —23  lb.  and  -f-io  lb.; 

(5)  +  x\h.    "      +ylh.; 

(6)  +  :x:lb.    ''      -x\h. 


13.  Denoting  motion  northward  by  the  positive  or  plus  (+) 
sign  and  motion  southward  by  the  negative  or  minus  (— )  sign, 
and  supposing  a  ship  to  start  on  the  equator  and  sail  as  indi- 
cated, tell  the  latitude  of  the  ship  in  both  amount  and  sign 
for  each  pair  of  sailings: 

(i)  -f-28°then  +  2°;  (5)  -h;c°  then  -10°; 

(2)  4-  2°    "     -18°;  (6)  -x°     "     -10°; 

(3)  -Hi2°     "     -12°;  (7)  +x°    "     -  y°; 

(4)  +12°     "     -24°;  (8)  -x°    "     -  y°. 

14.  A  boy  starts  work  with  no  money.  He  earns  50/ 
(4-50/)  and  spends  40/  (—40/).  How  much  money  has  he 
then  ? 

15.  If  a  man's  debts  be  indicated  by  writing  D  before 
their  amount  and  his  possessions  (assets)  by  P  before  their 
amount,  what  is  the  condition  of  a  man's  affairs  if  his  debts 
and  possessions  are  indicated  by  P  $1,200  and  D  $1,000? 
by  P  $75  and  D  $50?  D  $75  and  P  $60?  D  $300  and 
P  $,1000? 

16.  If  water  pushes  (buoys)  a  floating  body  upward  with 
a  force  of  18  lb.,  and  the  body's  weight  pulls  it  downward 
with  a  force  of  10  lb.,  the  two  forces  together  equal  what 
single  force  ? 

17.  If  a  man  was  born  40  b.  c.  and  died  45  a.  d.,  how  old 
was  he  when  he  died  ? 


4  First- Year  Mathematics 

i8.  Denoting  a  date  A.  d.  by  +  and  b.  c.  by  — ,  give  the 
length  of  time  between  these  pairs  of  dates: 

(i)  -  5  to  +io;     (3)  -  52  to  -50;     (5)  -  150  to  +   150; 
(2)  +16  to  +86;     (4)  —100  to  +50;     (6)  +1600  to  +1900. 

19.  Virgil  was  born  —70  and  died  —19;  how  old  was  he 
at  death  ? 

20.  The  first  Punic  War  lasted  from  —264  to  —241 ;  how 
long  did  it  last  ? 

21.  Egypt  was  a  Roman  Province  from  —30  to  +616; 
how  many  years  was  this  ? 

22.  Augustus  was  Emperor  of  Rome  from  —50  to  +14; 
how  many  years  was  he  Emperor  ? 

23.  What  will  denote  the  distance  and  direction  from  your 
school  house  to  your  home,  if  the  distance  and  direction  from 
your  home  to  your  school  house  are  denoted  by  +60  rd.  ? 
+  i^mi.  ?     +:x;  rd.  ?     —  80  rd.  ?     — i^mi.  ?     —ami.? 

24.  While  a  freight  train  is  moving  at  the  rate  of  10  mi.  an 
hour  toward  the  south  (  +  10  mi.  an  hr.)  a  brakeman  walks 
along  the  top  of  the  cars  toward  the  north  at  the  rate  of  4  mi. 
an  hour  (—4  mi.  an  hr.).  How  fast  and  in  what  direction 
does  the  brakeman  move  over  the  ground  ?  Answer  with  the 
aid  of  the  plus,  or  minus,  sign. 

25.  The  conductor  of  a  passenger  train  walks  from  the 
front  toward  the  rear  of  the  train  at  the  rate  of  3  mi.  an  hour 
while  the  train  is  running  at  the  rate  of  1 2  mi.  an  hour.  How 
fast  does  the  conductor  move  over  the  ground  ?  Answer 
with  the  aid  of  the  (  +  ),  or  (— ),  sign,  supposing  that  +  means 
toward  the  north  and  first,  that  the  train  is  running  north; 
then  second,  that  the  train  is  running  south. 

26.  What  would  the  sign  (— )  denote  if  the  sign  (  +  )  de- 
notes: (i)  above?  (2)  forward?  (3)  upward?  (4)  to  the 
right  ?   (5)  after  ?   (6)  east  ?   (7)  north  ?   (8)  possessions  ? 

From  these  problems  the  need  for  distinguishing  numbers 


Number  Generalized  5 

of  opposite  nature  is  evident.  It  is  clear  that  the  positive  and 
negative  signs  afford  a  convenient  means  of  making  this  dis- 
tinction. We  have  seen  also  that  letters  as  well  as  figures 
may  be  used  to  denote  numbers. 

A  number  having  a  positive  or  plus  (+)  sign  before  it  is 
called  a  positive  number. 

What  is  a  negative  number  ? 

The  positive  sign  (+)  need  not  always  be  written.  It  is 
generally  omitted  from  the  first  number  in  an  expression. 
The  negative  sign  is  never  omitted.  An  expression  like 
-\-x—a,  where  the  first  number,  x,  is  positive,  would  com- 
monly be  written  x—a,  and  is  read  "x  minus  a."  The  posi- 
tive sign  is  said  to  be  "understood"  in  this  case. 

It  is  now  necessary  to  learn  how  to  add,  subtract,  mul- 
tiply, and  divide  both  positive  and  negative  numbers. 


CHAPTER  II 
THE  OPERATIONS  APPLIED  TO  POSITIVE  NUMBERS 
§2.  Indicating  Arithmetical  Operations  Algebraically 

1.  A  boy  rides  on  his  bicycle  8  mi.  in  one  hour  and  5  mi. 
the  next  hour;  how  far  does  he  ride  in  the  two  hours  ? 

Note. — ^Write  the  sum  of  8  and  5,  not  13,  but  8  +  5.  The 
form  8  +  5  is  just  as  truly  a  sum  as  is  13.  It  will  sometimes 
be  desirable  to  refer  to  the  form  8  +  5  as  the  indicated  sum. 

2.  If  the  boy  rides  a  mi.  the  first,  and  7  mi.  the  second  hour, 
how  far  does  he  ride  in  the  two  hours  ? 

3.  A  boy  has  m  marbles  and  buys  p  more;  how  many  has 
he  then  ? 

4.  A  boy  had  18  marbles  and  lost  7 ;  how  many  had  he  then  ? 
Note. — Give  the  indicated  difference. 

5.  A  boy  had  m  marbles  and  lost  8  of  them;  how  many 
had  he  left  ? 

6.  A  boy  had  m  marbles  and  lost  n  of  them;  how  many 
had  he  left  ? 

7.  Show  the  sums  of  these  pairs  of  numbers  and  the  differ- 
ences, the  first  of  the  given  numbers  being  the  minuend,  and 
the  second,  the  subtrahend: 


(i)  X  and    7 ; 

(5) 

s  and   /; 

(9)  x  and  a; 

(2)  a-  -    12; 

(6) 

15    '*    n; 

(10)   d   "      c; 

(3)  y  "    10; 

(7) 

r    "     5; 

(11)   c    "      b; 

(4)  X    "      y; 

(8) 

a    "    x; 

(12)    t     "    m. 

8.  How  many  yards  of  cloth  are  12  yd.  and  10  yd.  ? 

9.  How  many  dozens  of  eggs  are  8  doz.  and  4  doz.  ? 

10.  How  many  12's  are  5  12's  and  4  12's? 

1 1 .  How  many  times  1 2  are  9X12  and  4X12? 

12.  How  many  half-dozens  are  8  half-dozens  and  6  half- 
dozens  ? 

6 


The  Operations  Applied  to  Positive  Numbers 


13.  How  many  times  6  are  8X6  and  6X6  ? 

14.  How  many  times  3  are  5X3  and  7X3  and  10X3  ? 

15.  How  many  times  x  are  2  times  x  and  3  times  x}  x 
and  x}  X  and  3  times  x ? 

Note. — It  is  customary  in  algebra  to  write  x+x,  x+x-{-x, 
x-{-x+x+x,  etc.,  thus  2x,  3^,  ^x,  and  to  read  them  "two  x" 
"three  x"  "four  x"  and  so  forth. 

But  X  times  x,  x  times  x  times  x,  and  x  times  x  times 
X  times  x,  or  using  the  dot,  which  stands  for  the  multipli- 
cation sign,  X  •  X,  X  •  X  •  X*  and  x  •  x  •  x  '  x,  etc.,  are  written 
x'^,  x^,  X*,  etc.,  and  are  read,  ":x;  square,"  "x  cube,"  ":x;  fourth 
power,"  etc.  x  square  may  be  read  "x  2nd  power  "  or  "  :x;  2nd ; " 
51?  cube  may  be  read  "rv  3rd  power,"  or'^x  ycd;^^  and,  "rv  fourth 
power"  may  be  called  "x  4th,"  etc. 

16.  What  would  be  the  written  form  of  "x  5th  ?"  ''x  6th  ?" 
":x;7th?"  "xioth?"  "xwth?"  What  would  be  the  meaning 
of  each  of  these  forms  ? 

Caution. — Notice  that  ^x  means  4  •  rv,  or  that  x  is  to  be 
used  as  an  addend  4  times,  while  x^ 
means  x-  x-  X'  x,  or  that  x  is  to  be 
used  as  a  factor  4  times,  and  similarly, 
for  the  other  forms,  as  ^x  and  x^,  and 
^x  and  x5,  etc.  -3x 

17.  Describe  briefly  the  meaning  Fig.  i 
of  factor  and  of  addend  as  they  are 
used  in  arithmetic. 

18.  What  is  the  sum  of  the  sides 
of  a  triangle  whose  sides  are  2X  ft., 
2X  ft.,  and  3^  ft.  long  ?  Express  the 
sum  as  a  certain  number  of  tiines  x. 

19.  What  is  the  sum  of  the  three 
sides  20,  5a,  and  6a  of  a  triangle  ? 

20.  A    lot   has  the    forni    of   an 
equal-sided    (equilateral)   triangle,    each    side    being    x  rd. 


„ ^ .^  -^  "" 

.    -y                 -JC.  22.  Wh; 

/'^                              /n  d.  square  of 

^— /  sq.  ft.  ?    a^ 


8  First-Year  Mathemniics 

long.  How  many  rods  of  fence  will  be  needed  to  enclose 
it? 

Def. — Any  figure  whose  sides  are  all  equal  is  called  an  equi- 
lateral figure,  a^  equilateral  triangle,  equilateral  pentagon,  etc. 
The  sum  of  all  the  sides  of  any  closed  figure  is  called  the 
perimeter  of  the  figure. 

21 .  In  Figure  3  what  part  of  the  perimeter  does  x-\-y  equal  ? 
What  is  the  whole  perimeter  ? 

22.  What  is  the  perimeter  of 

a  square  of  which  the  area  is  16 

sq.  ft.  ?    a'  sq.  ft.  ?    4a'  sq.  ft.  ? 

^  -^  ^^  sq.  ft.  ? 

^^'  ^  23.  What  is    the   area  of  a 

square  whosd  perimeter  is  20  ft.  ?    4X  ft.  ?    8a  ft.  ? 

24.  Draw  figures  to  illustrate  your  solutions  of  23. 

25.  The  dimensions  of  a  rectangle  are  a  and  b,  what  is  the 
perimeter?  The  half-perimeter?  The  sum  of  a  pair  of 
opposite  sides  ?    The  sum  of  the  other  pair  ? 

26.  What  is  the  area  of  the  rectangle  of  problem  25  ? 
Numbers   denoted    by   letters   are  literal  numbers.     The 

product  of  two  different  literal  numbers  as  x  and  y,  is  shown 
by  writing  the  letters,  or  factors,  side  by  side,  as  xy,  with  no 
sign  between.  We  are  famiUar  with  the  form  xXy  from 
arithmetic.     The  form  xy  is  most  used  in  algebra. 

We  have  already  seen,  p.  7,  that  the  product  of  x  by  x,  or 
of  a  by  a  by  a,  etc.,  is  written  x',  or  a^,  etc. 

27.  What  is  the  area  of  the  rectangle  that  is  8  in.  long  and 
5  in.  wide  ?  Of  the  same  length  and  4  in.  wide  ?  3^  in. 
wide  ?    6i  in.  wide  ? 

28.  What  is  the  area  of  a  rectangle  12  in.  long  and  of  the 
following  widths:  6  in.  ?  8 J  in.  ?  9^  in.  ?  lof  in.  ?  rv  in.  ? 
y  in.  ? 

29.  What  are  the  areas  of  rectangles  /  in.  long  and  of  the 
following  widths:  12  in.  ?  9  in.  ?  h  in.  ?  »  in.  ?  5C  in.  ?  a  in.  ? 


The  Operations  Applied  to  Positive  Numbers 


(lo)  a  by  ^*; 
(ii)  a^hy  X  ; 
(12)  ^  by  ^^ 


(1) 

iH-i 

(1)^ 

01 

30.  Give  the  areas  of  rectangles  of  width  w  and  of  the 
following  lengths:  8;  10;  12J;  x;  a;  I;  b;  z. 

31.  Write  the  products  of  the  following  pairs  of  factors: 
(i)  ahy  X  ;  (5)  ^    by  b^;  (9)  rv^  by  :v  ; 

(2)  b  by  c  ;  (6)  a^  by  a  ; 

(3)  bhy  b  ;  (7)  a'hya^; 

(4)  a  by  a';  (8)  x  by  a;3 ; 

32.  Indicate  the  area  of  a  rectangle  of  dimensions  a-\-b 
and  x+y. 

Note. — The    product    of    m+n    and    c+d    is    written 
(m+n)(c  +  d). 

33.  Express  in  terms  of  its  base  and  altitude  the  area  of 
the   rectangle  (i)  of  Fig.  4;  of  (2); 

of  (3);  of  (4).  ^ 

34.  How  then  may  you  express 
(m+n)  (c+d)  by  using   the  truth    ^f^ 
that  any  whole  equals  the   sum  of 
all  its  parts? 

35.  State   in  words  the  value  of 
(x+y)  (a+b);   (x+y)  (m+n);  (a+x)  (b  +  y);   (r+s)  (a+x); 
(2a+3b)(x+y);  (a+4x)(sb  +  y). 

36.  Show  by  a  figure  the  value  of  (a+b)(a+b),  or  (a+b)'; 
of  (c+d)'';  of  (x+y)';  of  (m+n)';  of  (2a+d)';  of  (4X+2y)'. 

37.  Each  of  the  following  expressions  is   the  product  of 
what  two  equal  numbers  ? 

(i)  a'  +  2ax+x';  (3)  k'  +  2kb+b';  (5)  x'  +  6x+  9; 

(2)  b'  +  2bc +c';  (4)  s'  +  2st  +  t';  (6)  c' +8c +16. 

38.  The  base  of  a  rectangle  is  8  yd.  and  the  area  is  40 
sq.  yd.     What  is  the  altitude  ? 

39.  What  is  the  altitude  of  a  rectangle  having — : 

a  base =8  in.  and  an  area =3  2  sq.  in.  ? 
"  =8  in.  "  "  =i6sq.  in.? 
"  =8  in.  "  "  =i2sq.  in.  ? 
"     =5  ft.      "        "       =7isq.  ft.? 


tn 


Fig.  4 


TO 


First-  Year  Mathematics 


40.  What  is  the  base  of  a  rectangle  having — 

an  altitude =9  ft.  and  an  area  =  27  sq.  ft.  ? 
=9  ft.  "  "  -18  sq.ft.? 
=9  ft.  "  "  =15  sq.ft.? 
=9  ft.  "  "  =12  sq.ft.? 
=9  ft.       "       "      =  6  sq.  ft.  ? 

41.  What  is  the  other  dimension  of  a  triangle  having — 

a  base  =  6  ft.  and  an  area  =  24  sq.  ft.  ? 
"  =6  ft.  "  "  =12  sq.ft.? 
"  =6  ft.  "  "  =  9  sq.ft.? 
"       =6  ft.     "        "       =  3  sq.ft.? 

an  altitude =4  yd.  and  an  area  =  16  sq.  yd.  ? 


=4  yd. 

'     =  8  sq.  yd.  ? 

=4  yd. 

'    =  4  sq.  yd.  ? 

=aft. 

'     =  a  sq.   ft.  ? 

=  hTd. 

'     =  hi  sq.  rd.  ? 

=b  in. 

'     =ab  sq.  in.  ? 

=b  in. 

'     =bysq.  in.  ? 

=c  in. 

'     =a    sq.  in.  ? 

=a  in. 

'    =b    sq.  in.  ? 

The  quotient  of  x  divided  by  y  is  written  -,  or  x-v-y,  and 

X 

read  '^x  divided  hy  y."     -  is  also  read  "x  over  y." 

42.  Write  the  quotient  of  the  first  of  these  numbers  divided 
by  the  second: 

(i)  w  and  w;  (6)  a+b  dindc+d;  (11)  a'  —  b''*SLnda-{-b 

(2)  c     "    n;  (7)  4X       "       3^:  (12)  a^"-*'    "     a-6 

(3)  a     "    b;  (8)  3^       "       4j;  (13)  ^'    "     «  +  & 

(4)  b     "   a;  (9)  a+6   "          c;  (14)  (a+&)^    "     a+b 

(5)  :r     "   y;  (10)  a         "   :v+:y;  (15)  (a-by    "     a-&. 

*  Note,  a'  —  6*  is  read  ' ' a  square  minus  b  square ;"(a—by 
is  read  "the  square  oi  a—b,"  and  (a+by  is  read  "the  square 
of  a+b." 


CHAPTER  III 

THE  ARITHMETICAL  OPERATIONS  WITH  NUMBERS   REPRE- 
SENTED BY  LINES 

§3.     Operations  with  Lines 

Sums,  differences,  products,  and  quotients  may  be  con- 
structed. For  notebook  work  in  construction  a  ruler  and  a 
pair  of  compasses  are  needed.  For  blackboard  work  the  com- 
passes may  be  replaced  by  chalk  and  string. 

A S: B 

C ^ JD 


a 


l-L-4- 


<--  -  -  -  -  ^+^ —  — > 

Fig.  5. 

1.  To  construct  the  sum  of  two  lines,  for  example  a  and  b 

(Fig.  5)- 

Construction. — Draw  a  straight  Une,  as  OL,  longer  than  a  and 
b  together  when  placed  end  to  end.  Open  the  compasses  so  that 
the  points  will  be  a  distance  apart  equal  to  a.  With  the  pin-point 
at  O  mark  a  short  arc  across  OL  at  E,  with  the  pencil  point. 

Then  with  the  distance,  b,  between  the  points  of  the  com- 
passes and  with  the  pin-point  on  E,  mark  a  second  arc,  as  at 
F,  across  OL.  What  line  then  has  the  length  equal  to  a-f-6? 
Would  the  sum  be  the  same  if  the  lines  a  and  b  were  added 
in  the  reverse  order  ? 

2.  Draw  a  pair  of  lines  in  your  notebook  and  on  another 
line  construct  their  sum. 

Remark. — The  sum  of  two  lines  of  different  lengths  on 
the  blackboard  may  be  constructed  with  crayon  and  string. 

3.  Draw  lines  to  represent  2a  and  36  and  show  how  to 
construct  2a+7,b. 


12  First-Year  Mathematics 

4.  Construct  w  +  2c?;  x-\-y+z;  x+2y+2z;  2{a  +  b); 
S(a+b);  30+36. 

5.  Compare  the  sums  20  +  26  and   2(0  +  6);  3(0+6)   and 

6.  To  construct  the  difference  of  two  lines,  proceed  as  with 

PL the  sum  until  the  subtrahend 

L  line    is    to   be   marked    off. 

^--|.---^  Instead  of  marking  D  on 

O ^ 1 beyond  E  on  OL  allow  D  to 

< CL >  fall  on  the  other  side  of  E, 

^^°-  ^  i.  e.,   back    on    the   minuend 

line.     Point  out  in  the  adjoining  figure  (Fig.  6),  the  line  0—6. 

7.  Construct  the  dififerences  of  these  pairs  of  lines,  the  first 
being  the  minuend  line: 

(i)  m  and  n,  (m>n);*  (5)  x  and  y,  (x=y); 

(2)  fl     "    x,(a>x);  (6)  c    "    d,  (c=d); 

(3)  c     "     d,(c  >d);  (7)  2X  and  2x; 

(4)  />    "     x,(p  >x);  (8)  3;  and  J. 
*Note:    o>6  indicates  that  o  is  greater  than  6. 

Notice  that  if  the  subtrahend  line  is  the  longer  the  point 
D  will  fall  beyond  O  (i.  e.,  to  the  left  on  the  figure).  When- 
ever the  difference  (OD)  is  measured,  or  extends,  toward  the 
right  of  O  (or  along  the  minuend  line)  it  will  be  positive ;  when 
it  extends  toward  the  left  from  O,  i.  e.,  on  the  minuend  line 
prolonged  through  O,  it  will  be  negative. 

8.  Show  how  to  subtract  4  from  3;  8  from  5;  6  from  3; 
7  from  2;  9  from  i;  3  from  o;  and  state  both  the  magnitude 
and  the  sign  (direction)  of  the  differences. 

g.  Construct  the  following  differences;  the  first  number 
denoting  the  minuend  line: 

(i)  3:x;  and    x;  (4)  20  and  26;  (7)  o  and  40; 

(2)  2X    "    ^x;  (5)  60    "    40;  (8)  4wand6w. 

(3)  X    "  5^;  (6)  3^  "  4a; 

10.  Construct  these  expressions: 
(i)  a+x-y;  (3)  2(a-x+y);  (5)  2(o-6  +  2<;); 

(2)  a  +  2X—2y;  (4)  2(0  +  6  — 2c);  (6)  30  —  26—0. 


Operations  with  Numbers  Represented  hy  Lines        13 


§  4.     Sums  and  Differences  of  Angles 

1.  Two  angles,  as  x  and  y,  Fig.  7,  may  be  added  by  placing 
together  one  side,  as  AC  of  angle  y,  along  a  side,  as  AC  of 
the  other  angle,  x. 
If  AD  lies  on  the 
side  of  AC  opposite 
to  AB,  the  angle  be- 
tween AB  and  AD 
will  be  the  angle 
x-\-y. 

2.  If  the  angle,  y, 
is  turned  over  AC 
as  a  hinge,  making 

AD  to  lie  along  the  A  -3 

dotted  Hne  AD',  the  ^'^-  ^ 

angle  between  AD'  and  AB  is  the  diflference,  x—y. 

3.  Draw  two  angles,  a  and  h,  on  the  black  board,  and 
show  how  to  obtain  their  sum  without  measuring  the  angles. 
Show  how  to  obtain  their  difference  without  measuring. 

4.  Show  the  sum  and  the  difference  of  two  angles  by  fold- 
ing or  cutting  paper. 


§  5.     Products  as  Rectangles 

The  product  of  two  numbers  may  be  constructed  as  follows : 
^  Let   the   two   numbers   be   repre- 

sented by  lines,  as  x  and  y,  in  Fig.  8. 
Construction:  At  the  end  A  of 
AB  {=x)  draw  a  perpendicular  Une 
AD  and  make  it  equal  to  y  in  length. 
Through  D  draw  DC  parallel  to  AB 
and  through  B  draw  BC  parallel  to 
AD.  The  rectangle  ABCD  represents 
the  product  of  x  and  y.  The  rec- 
tangle denotes  the  product,  in  the  sense  that  it  contains  as 


14  First-Year  Mathematics 

many  square  units  of  area  as  there  are  units  in  the  number 
xy. 

1.  Show  the  product  of  two  lines  of  given  lengths  on  the 
blackboard. 

2.  Draw  the  product  of  one  or  more  pairs  of  lines  in  your 
notebook. 

3.  Construct  the  product  of  two  lines  2x  and  x,    of  ^x 
and  2x. 


CHAPTER    IV 


ADDITION,  SUBTRACTION,    AND  MULTIPLICATION   OF   POSI- 
TIVE AND  NEGATIVE  WHOLE  NUMBERS 

§  6.     Adding  Positive  and  Negative  Whole  Numbers 

I .  Denoting  distances  traveled  northward  by  positive  num- 
bers, and  distances  traveled  southward  by  negative  numbers, 
find  for  each  of  the  following  cases  the  distances  and  the  direc- 
tion of  the  stopping  point  from  the  starting  point.  When  the 
stopping  point  is  north  of  the  starting  point  mark  the  result 
+  ;   when  south,  mark  the  result  — . 

An  automobile  goes: 


(i)  +15  mi.,  then  —10  mi. 

(2)  -I-15  mi.,    "     —14  mi. 

(3)  +15  mi.,    "     -20  mi. 

(4)  +25  mi.,    "     -35  mi. 

(5)  —18  mi.,    "     -I-24  mi. 


(6)  —12  mi.,  then  —10  mi.; 

(7)  —20  mi.,     "    -I-15  mi.; 

(8)  —15  mi.,     "    -1-22  mi.; 
(8)  —20  mi.,     "    -|-2i  mi. ; 

(10)  —22  mi.,     "    4-22  mi. 


2.  In  the  following  problems  the  numbers  indicate  distances 
traveled  northward,  if  negative;  and  southward,  if  positive. 
The  sum  in  all  cases  must  denote  the  distance  and  direction  of 
the  stopping  point  from  the  starting  point.  Write  the  sums 
with  their  proper  signs: 


(I) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

+15 

-15 

+  15 

-15 

+38 

-38 

+38 

-38 

+  8 

-  8 

-  8 

+  8 

+  19 

-19 

+  19 

+  19 

(9) 

(10) 

(11) 

(12) 

(13) 

+  4 

-  4 

+  11 

-  4 

+  12 

-1-26 

-26 

-26 

-1-26 

—  12. 

3.  Examine  (i),  (2),  (5),  (6),  (10)  of  problem  2  and  make 
a  rule  for  adding  two  numbers  having  like  signs. 

IS 


1 6  First-Year  Mathematics 

4.  From  (3),  (4),  (8),  (11),  (12),  and  (13)  of  problem  2 
.make  a  rule  for  adding  two  numbers  having  unlike  signs. 

Sums,  with  their  proper  signs,  of  positive  and  negative 
numbers,  are  called  algebraic  sums.  The  sums  and  diflfer- 
ences  of  numbers  regardless  of  sign,  are  called  arithmetical 
sums  and  differences. 

SUMMARY 

The  algebraic  sum  of  two  numbers  with  like  signs  is  their 
arithmetical  sum,  with  the  common  sign  prefixed. 

The  algebraic  sum  of  two  numbers  with  unlike  signs  is 
their  arithmetical  difference,  with  the  sign  of  the  larger  num- 
ber prefixed. 

5.  In  the  following  problems  the  positive  numbers  indicate 
gains  and  the  negative  numbers  indicate  losses.  The  sums 
indicate  the  net  change  in  the  man's  capital,  and  whether  the 
net  change  is  an  increase  or  a  decrease.  Find  the  sums  and 
tell  their  meaning : 


(I) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

+50 

+  35 

-45 

+  75 

-236 

+Sx 

—  14a 

+25 

-38 

—  20 

+  13 

+  780 

-6x 

—46a 

-18 

+  24 

+  60 

-86 

-  95 

-4X 

+  77a 

-  6 

-15 

+  55 

+  8 

+  45 

+  7^ 

-  5a 

6.  State  a  way  of  adding  any  number  of  positive  and 
negative  numbers. 

7.  A  force  of  12  lb.  pulling  toward  the  right  (  +  12  lb.) 

together  with  a  force  of  9  lb. 
pulUng  toward  the  left  give  a 
combined  pull  equal  to  what 
force  ? 

U  jj         8.  What  single  force  has 

Fig.  9  the    same   effect   in   pulling 

the  ring  R  as  the  following  pairs  of  forces  acting  together  ? 


0" 


Positive  and  Negative  Whole  Numbers  17 


(l) 

+  12  lb. 

and 

-  8  1b.; 

(9)   -16  lb. 

and 

-  8  1b. 

(2) 

-12  lb. 

+  8  1b.; 

(10)   +  3  lb. 

<( 

-12  lb. 

(3) 

-10  lb. 

+  10  lb.; 

(11)   +  x\h. 

It 

+  :yib. 

(4) 

-16  lb. 

+  13  lb.; 

(12)   +  x\h. 

(I 

-  ylh. 

(5) 

+  14  lb. 

-171b.; 

(13)   -  X  lb. 

(e 

+  ylh. 

(6) 

+  9  lb. 

—  20  lb.; 

(14)   -  X  lb. 

(I 

-  ylh. 

(7) 

+  11  lb. 

+  15  lb.; 

(15)  +  :x;lb. 

ii 

-  ^clb. 

(8) 

-16  lb. 

+13  lb.; 

(16)  -2^  lb. 

<( 

+  xlb. 

9.  A  man  draws  a  bucket  of  brick,  weighing  60  lb.,  to  a 
house  top  by  pulling  on  a  rope  which  runs  over  a  pulley,  with 
a  force  of  65  lb.  What  single  force  equals  the  sum  of  the  two 
forces  acting  on  the  handle  of  the  bucket  ? 

10.  A  balloon  pulls  upward  on  a  stone,  weighing  6  oz., 
with  a  force  of  8  oz.     What  is  the  sum  of  the  forces  ? 

11.  A  piece  of  iron  weighing  18  lb.,  when  placed  under 
water,  is  pushed  (buoyed)  upward  with  a  force  of  2^  pounds. 
What  is  the  sum  (combined  effect)  of  the  two  forces  together  ? 

12.  An  elevator  starts  at  a  certain  floor,  goes  up  65  ft., 
down  91  ft.,  up  52  ft.,  down  13  ft.,  and  up  65  ft.,  and  stops. 
How  far  and  in  what  direction  is  the  stopping  from  the  starting 
point  ?     Give  your  answer  in  the  form  of  an  algebraic  sum. 

13.  A  vessel  starts  in  latitude  +20°,  it  sails  +13°  in  lati- 
tude, then  —60°,  then +40°,  then  —10°.  What  is  its  latitude 
after  the  sailings  ?  What  is  the  latitude  of  a  ship  starting  in 
latitude  —50°  after  these  changes  of  latitude:  +10°,  —5°, 
+  18°,  -f,  +38°,  -12°,  +60°? 

14.  A  boatman  rows  at  a  rate  that  would  carry  him  3 
miles  an  hour  through  still  water,  down  a  river  whose  current 
is  2  mi.  an  hour.  What  is  his  rate  per  hour  ?  What  would 
be  his  rate  per  hour,  if  he  rows  up  the  river  ? 

§  7.     Subtracting  Positive  and  Negative  Whole  Numbers 

I.  In  this  problem  positive  numbers  indicate  the  readings 
above  zero  and  negative  numbers,  readings  below  zero.    The 


(4) 

(5) 

-   f 

+  i' 

+  32° 

-28' 

(6) 

(7) 

0° 

-8° 

+  78° 

+8° 

(8) 

(9) 

+6x° 

-4a° 

-3^° 

-9a° 

(10) 

-  2/ 

+6ov° 


18  First-Year  Mathematics 

difference  means  the  number  of  degrees  the  top  of  the  mercury 
column  must  rise,  or  fall,  to  change  from  the  second  reading 
to  the  first.  If  the  change  is  a  rise,  mark  it  positive  (+),  if 
a  fall,  mark  it  negative  (— ). 

(i)         (2)  (3) 

First  reading:       +68°     -98°     -30° 
Second  reading :  +42°     —18°     +65° 

First  reading : 
Second  reading: 

Define  minuend.     Define  subtrahend. 

2.  By  the  summary,  p.  16,  find  the  sums  in  the  following 
problems  and  compare  the  exercises  and  your  results  with 
those  of  the  Uke  numbered  exercises  of  problem  i : 

(i)       (2)       (3)       (4)-    (5)       (6)      (7)       (8)       (9)       (10) 
+  68    —98    —30    —7+1         o    —8    +6x    —4a     —  2y 
—42    +18    —65    —32    +28    —78    —8    +7,%    +9a     —6oy 

3.  Show  by  comparing  the  problems  of  i  and  2  that  the 
difference  of  any  two  numbers  can  be  found  by  changing  the 
sign  of  the  subtrahend  and  then  adding. 

4.  Find  the  differences  of  the  following: 

(I)  (2)         (3)  (4)  (5)  (6)  (7) 

+  19       —60       —75       +  8       —3a      +i2>x      —i2y 
—  ID      —25       +25       —16      —2a      —  dx      —  Ty 


(8) 

(9) 

(10) 

(11) 

+a 

+3^-18 

+  a+26—  c 

—  i2a+3:x; 

~h 

+  2:x;+  6 

+3a-56+3C 

+  ']a—2x 

§  8.     Multiplication  of  Positive  and  Negative  Whole  Numbers 

Suppose  the  short  spaces  on  the  line  east-west,  E  W,  repre- 
sent a  mile. 


Positive  and  Negative  Whole  Numbers  19 

I.  Show  what  space  starting  from  o  in  each  case  repre- 
sents +  2  mi.;    +3  mi.;  +  5  mi.;+8  mi.;   +10  mi. 


u/     ■  ■ ,  ■  . .  ^ 

Fio.  10 


2.  Starting  again  from  o,  show  the  space  that  represents 
— I  mi.;  —2  mi.;    —5  mi.;    —7  mi.;    —10  mi. 

3.  Show  the  spaces  from  o  a  man  goes  if  he  travels  +2  mi. 
a  day  for  i  day;  2  days;  3  days;  5  days. 

4.  What  is  2  times  —2  mi.?    3  times  —2  mi.?    4  times 

—  2  mi.  ?     5  times  —2  mi.  ? 

5.  What  is  5  times  —4  mi.  ?  — 10  mi.  ?  — 20  mi.  ?  —25  mi.  ? 

—  100  mi.  ? 

6.  What  is   16   times    —2   mi.?     —5   mi.?     —10   mi.? 

—  2omi.  ?     —100  mi.  ? 

7.  What  is  5oX(-2)?  25X(-io)?  4oX(-i2)? 
8X(-i2o)?  4X(-a)?   ioX(-^)? 

8.  The  value  of  a  man's  property  changes  by  +1000  a 
year.  How  much  does  it  change  in  2  yr.  ?  5  yr.  ?  8  yr.  ?  10 
yr.  ?  In  each  case  tell  whether  the  change  is  an  increase,  or 
a  decrease. 

9.  If  the  value  of  a  man's  property  changes  by  —$500  a 
year,  how  much  does  it  change  in  3  yr,  ?  5  yr.  ?  7  yr.  ?  8  yr.  ? 
10  yr.  ?  12  yr.  ?  In  each  case  tell  whether  the  change  is  an 
increase,  or  a  decrease. 

10.  How  much  and  in  what  way  does  a  man's  property 
change  in  12  yr.  at  the  rate  of  $50  a  yr.  ?  — $100  a  yr.  ?  +$400 
a  yr.  ?   +$1,800  a  yr.  ?    —$2,000  a  year  ? 

11.  How  much  and  in  what  direction  does  the  height  of  a 
mercury  column  of  a  thermometer  change  in  6  hours  at  the 
rate  of  +10°  an  hr.  ?  -8°  an  hr.  ?  -f  an  hr.  ?  +5^°  an 
hr.  ?   — 3^°  an  hr.  ?    +2^°  an  hr.  ?    —a°  an  hour  ? 


20 


First-Year  Mathematics 


12.  State  a  way  of  multiplying  a  negative  number  by  an 
arithmetical  number. 

The  bar,  AB,  is  balanced  on  the  middle  peg,  marked  O. 
There  are  5  equally  spaced  pegs  on  either  side  of  O.  Equal 
weights  provided  with  hooks  may  be  hung  on  the  pegs.  A 
smooth  rod,  CD,  fitted  with  a  pulley,  P,  over  which  a  cord  passes, 
is  held  in  such  position  that  weights  may  be  hung  at  one  end 
of  the  cord.     The  other  end  of  the  cord  may  be  hooked  to 


Fig.  II 


any  of  the  pegs.  By  moving  the  sUding  pulley  along,  the 
force  of  the  weight  may  be  made  to  pull  vertically  upon  the 
peg.  Notice  that  with  this  apparatus  we  may  use  forces  acting 
either  upward  or  downward,  on  pegs  either  to  the  right  or 
left  of  the  center.  The  small  weights  are  all  equal  and  the 
spaces  between  the  pegs  are  all  equal. 

Call  a  weight  w  oz.,  and  a  space  x  inches. 

13.  How  shall  we  distinguish  distances  measured  to  the 
right  of  O  from  those  measured  to  the  left  ? 

14.  How  shall  we  distinguish  forces  pulUng  upward  from 
those  pulling  downward  ? 

15.  If  the  bar  is  balanced  and  a  weight  is  then  hung  at 
+  1,  what  will  occur?  What  will  occur  if  the  weight,  w,  is 
hung  on  peg +2?     +3?   +4?  +5?  -2?  —3?   —4?  —5? 

16.  Suppose  that  no  weights  are  hung  to  the  pegs,  but  that 


Positive  and  Negative  Whole  Numbers  21 

a  weight  hangs  from  the  end  of  the  cord  and  pulls  upward  on 
peg  + 1 ;  what  will  occur  ?  What  will  occur  if  the  weight  pulls 
upward  on  peg  +3  ?    +5?    — i  ?    —2?    —3?    —5? 

17.  Downward  forces  on  any  peg  to  the  right  of  O  have  a 
tendency  to  turn  the  bar  around  O  in  what  direction  ?  What 
turning  tendency  do  downward  forces  have  on  any  peg  to  the 
left  ?  Upward  forces  on  positive  pegs  (pegs  to  the  right)  ? 
Upward  forces  on  negative  pegs  ? 

18.  What  two  turning  tendencies  must  we  deal  with  ? 
Note. — Imagine  a  watch  laid  down  on  the  drawing,  face 

up,  with  the  hand  post  just  over  O.  When  the  bar  turns,  or 
tends  to  turn,  around  wth  the  hands  of  the  watch,  the  bar  is 
said  to  have  a  negative,  or  minus  (— ),  turning  tendency.  If 
the  tendency  is  against  (opposite  to)  that  of  the  watch  hands, 
it  is  a  positive,  or  plus  (+),  tendency. 

Definition. — The  distance  from  the  turning  point,  O,  to 
the  point  where  the  force  or  weight  acts  is  called  the  lever 
arm,  or  arm  of  the  force. 

The  turning  tendency  of  any  weight  or  force  is  the  product 
of  the  number  of  units  in  the  force  by  the  number  of  units  in 
the  arm  of  the  force.  Or  T=fXa,  where  T,  is  the  turning 
tendency,  /  the  force,  and  a  the  arm. 

19.  What  is  the  turning  tendency  of  a  weight,  or  force, 
—  2ze;  (2W  pulling  downward)  on  the  peg  +12?  Of  a  force 
-\-w  on  the  peg  +2  ? 

Solution. — (i)The  turning  tendency,  T,  is  (  — 2w)  X(  +  i^); 
since  the  force  is  —2W  and  the  arm  is  +1^,  x,  or  +x.  qwXx 
=  2wx,  and  as  the  bar  tends  to  turn  with  the  hands  of  the 
watch,  T=—2wx.  (2)  T=(+w)X(  +  2x)  =  +2wx  (+  for 
what  reason?). 

20.  What  is  the  turning  tendency  of  +w  at  +3  ?  +2w  at 
+  2  ?  +3W  at  +4  ?  —w  at  +3  ?  —2w  at  +4  ?  — 3^  at  — i  ? 
— 2W  at  —3  ?  —2W  Sit  —4  ?  +2W  at  —2  ?  +3W  at  —2  ?  +2W 
at  —4? 


23  First-Year  Mathematics 

21.  What  sign  (direction)  has  the  turning  tendency  of  a 
plus  force  with  a  plus  arm  ?  A  plus  force  with  a  minus  arm  ? 
A  minus  force  with  a  plus  arm  ?  A  minus  force  with  a  minus 
arm? 

22.  If  3  (—4)  means  that  —4  is  to  be  measured,  or  laid 
off,  3  times  from  o  in  the  negative  direction,  what  is  the  value 


Fig.  12 

of  the  product,  3 (—4),  or,  what  is  the  same  thing,  (+3)(— 4)? 
The  product  means  the  distance  and  direction  this  leaves  us 
from  o,  evidently  —12. 

23.  Show  on  a  figure  the  meaning  and  value  of  the  fol- 
lowing products: 

(1)  2(+3);        (3)  3(-2);         (5)  3(-5);        (7)  ^(+6); 

(2)  2(-3);  (4)  3(  +  2);  (6)  3(  +  5);  (8)  2(-6). 

24.  Show  on  a  figure  the  product  (— 3)(— 4). 

Note.  (—3) (—4)  means  that  four  is  to  be  laid  off  3 
times  in  the  direction  opposite  to  the  direction  of  —4,  i.  e., 
in  the  positive  direction. 

25.  Interpret  these  products  on  the  same  principle. 


(i)  (-2)(-3);  (4)  (-2)(-8) 

(2)  (+3)(-2);  (5)  {-i){-s) 

(3)  (-2)(+4);  (6)  {+z){-s) 


(7)  (-2)(+5); 

(8)  (-2)(-5); 

(9)  (-3)(  +  6). 


26.  Show  on  Fig.  13  the  value  of  2  times    +a,    2 (—a); 
(-2)(+a);   (-2)(-a);   (-3)(-a);   (-3)(+^);  (-4)(-a). 


Fig.  13 


Positive  and  Negative  Whole  Numbers 


23 


27.  Interpret  the  meanings  of: 

(i)  (+3)(+«);  (5)  (+a)(+6) 

(2)  (+3)(-«);  (6)  (+a)(-6) 

(3)  (-3)(+^);  (7)  (-«)(+*) 

(4)  (-3)(-^);  (8)  {-o){-h) 


(9)  (+c)(-(i) 

(10)  (+c)(+cO 

(11)  (-0(+^ 

(12)  (-.)(-(?) 


SUMMARY 


In  digits: 
(+3)(+4)  =  +  i2; 
(+3)(-4)  =  -i2; 
(-3)(+4)  =  -i2; 
(-3)(-4)  =  +  i2; 


In  letters: 
(+a)(  +  6)  =  +fl6; 
{+a){-h)  =  -ah; 
(-a)(  +  b)  =  -ab; 
(-a)(-b)  =  +ab. 


28.  Examine  the  eight  products  of  the  summary  and  make 
a  rule  for  obtaining  the  algebraic  sign  of  a  product  of  two 
numbers  from  the  signs  of  the  factors. 

Compare  your  rule  with  this: 

§  9.     Law  of  Signs  for  Multiplication 

//  two  factors  have  the  same  sign,  their  product  is  positive; 
and  if  two  factors  have  unlike  signs,  their  product  is  negative. 


CHAPTER  V 
OPERATIONS  ON  FRACTIONAL  NUMBERS  GENERALIZED 

The  operations  of  addition,  subtraction,  multiplication, 
and  division  must  frequently  be  used  upon  fractions. 

§  10.     Unit-Fractions — Review-Problems 

1 .  What  is  the  sum  of  ^  and  ^  ?  -^  and  ^  ?  ^  and  ^  ?  tV 
and  xV  ? 

2.  What  is  the  sum  of  ^  and  ^  ?    ^  and  ^  ?   ^-^  and  ^  ? 

Explanation  of  ^+\:  |+i=-^+— =Z^  . 

2.7     2.7      7-2 

In  the  same  way  write  out  the  work  of  all  the  problems 

I  and  2. 

3.  Examine  your  results,  compare  them  with  the  numbers 
in  the  given  problem  and  make  a  rule  for  quickly  adding 
fractions  whose  numerators  are  i. 

Definition. — Fractions  having  i  for  numerators  are  called 
unit  fractions,  or  fractional  units. 

4.  Apply  your  rule  to  these  sums: 

(2)m;  ^^^^^-'  ^'^-^~y' 

(4)  Hi;  (6)-.-+^'  (^\.+m- 

5.  What  is  the  difference  of  -^  and  ^  ?  i  and  ^  ?  ^  and  ^  ? 
■J-  and  i  ?  ^  and  -^?  |  and  yV  ? 

Explanation  ofi-^:  i-|=-^^ ^  =  ^^^^  . 

^  ^     '     ^     '     5-7     5-7      5-7 

Note. — ^Write  out  the  work  of  all  the  parts  of  problems  5 

as  is  done  in  this  explanation. 

6.  Examine  your  results  and  make  a  rule  for  quickly  find- 
ing the  difference  of  unit  fractions. 

24 


operations  on  Fractional  Numbers  Generalized         25 

7.  Apply  your  rule  to  finding  these  differences: 
(2)i-^;  ^4)^--,  (6)--^,  (8)7--. 

(3)^4^     (s^H^     ^^^rr     ^'^-a-l 

8.  Show  that  the  addition  and  the  subtraction  of  unit  frac- 
tions may  be  indicated  thus : 

9.  Solve  the  following  problems: 

(i)  i±|;  /'r^^4-^.  M^M^-  U\^M^ 

(2)i±TV,  ^^^-^~y'  ^^^~d^~c'  ^^^"^^ 

§11.     Fractions  Having  the  Same  Numerator 

1.  Add  f  and  f . 

Solution:  |+f =3(1+1)  =3^y^=3  •  il  =  i^  • 

2.  Subtract  f  from  f . 

Solution:  f-f =3(-i-x)=3l=i=^=/^  . 

7-4     7*4 

3.  Solve  the  following  exercises: 

(2)f±|;  (5)-±-,  (7)^±-,  (9)^±, 

(4)f±A;        ^'^^^^'       ^'^J=^^'        ^^°^^±^- 

§  12.     Addition  and  Subtraction  of  Fractions  in  General 

I .  Add  f  and  | . 

c  .  .  ,     5-8 , 6-7     5-8+6-7 

Solution:  |-|-i=5_^  +  — 1=? 


6-8    6-8  6 


2.  Subtract  4  from  f . 


26  First- Year  Mathematics 

Solution:  ^-A=lll-il9^T^-^'9  , 
7-9    7-9         7-9 

3.  Add  -J    and   -^^   where  n^    (read   "»   one"),    ii   ("d 

one")  stand  for  a  first  numerator  and  a  first  denominator 
«a  (read  "n  two")  and  (i,  (read  "d  two")  stand  for  a 
second  numerator  and  a  second  denominator,  respectively. 

ft  fl 

4.  Subtract  -^  from  -^  .     The  numbers  n^,  dj,  n,,  and  rf, 

a,  a, 

indicate  the  same  as  in  problem  3. 

ft  ft 

Note. — Observe  that  the  subtraction  of   ^  from  -^  is 

a,  di 

ft      fl 
indicated  thus:  -^—-j-- 
d^     ttj 

5.  Make  a  rule  for  finding  the  sum,  or  the  diflference,  of 
any  two  fractions. 

6.  Write,  by  the  rule  just  made,  the  values  of  these  indicated 
sums  and  differences : 

W    UA;  (8)f    J  (..)^±2:; 

(*'  A+A;  :    ,  3-  ;• 

(5)A±A;  <'°)i±^;  ('"'jiy- 

(6)   xVif  >  ,      .    m      X  ,      ^    X       C 


§  13.     Multiplication  and  Division  of  Fractions  in  General 

1.  Multiply  f  by  4;   by  8;   by  12;   by  40;   by  5;   by  10; 
by  25;  by  200;  by  a. 

2.  Multiply  I  by  f ;   by  |;   by  |;   by  f ;   by  |;    by  ^  ; 

u     *     u     ^ 

by  - ;  by  -  . 
3  3 


Oi>eratwns  on  Fractional  Numbers  Generalized 


27 


3.  Solve  the  following: 

(i) 
(2) 

iXf 

(7) 

a     c 

(3) 

|X| 

(4) 

ixf 

(8) 

y    d 

(5) 

1X1 

(6) 

«x^ 

;            (9) 

y- 

5     7 

t     q 

,    s,  a^     aa 

4.  Make  a  rule  for  multiplying  any  two  fractions  together. 

5.  Apply  your  rule  to  these  indicated  products:     (A  dot, 
thus  (•),  also  denotes  multiplication;  as  4  •  4  =  20.) 

3 


(2)    f-f 


(3) 
(4) 
(5) 
(6) 
(7) 


i-i 

.7   .  8. 


9       7 


(8)    ?.|; 


:v     10 


(10) 


X   y 
y    X 


(12) 

(13) 

i' 

i; 

(14) 

i- 

i; 

(15) 

¥- 

•tV 

(16) 


a    I 


6.  Find  the  values  of  these  products: 

X    s 


(i)   i't-h 
(2)   f-f'f; 


(4)*-^r 


(3) 


(5) 


a    c    ^ 


(6) 


(7) 


«i     Wa     W3  _ 


di    di 


^3' 


Oi    a,    Oj 
61    bi    b. 


b    d    y' 

7.  Make  a  rule  for  multiplying  any  three  fractions  together. 

8.  Change  your  rule  to  make  it  apply  to  the  product  of  4 
fractions ;  of  n  fractions. 

9.  What  is  the  simplest  form  of  the  product  in  each  case 
of  problem  5  ? 

10.  By  examining  the  products  of  problem  5,  tell  without 

actually  dividing,  how  many  times  f  is  contained  in  i;   f  in 

oc 
I ;  f  in  I ;  f  in  i ;  f  in  i ;  |^  in  i ;   —  in  i ;  2  in  i ;  12  in  i ; 


28  First- Year  Mathematics 

.  .  .  ,  .  g  .  a  .  b  . 

fmi;ami;  ^iiiij  fmi;  rini;  -ini. 

11.  Give  a  quick  way  of  finding  how  many  times  any 
fraction,  or  any  whole  number,  is  contained  in  i. 

Principle  I. — The  reciprocal  of  a  number  shows  how  many 
times  it  is  contained  in  i. 

12.  How  many  times  are  these  numbers  contained  in  i  ? 

(2)f?     (4)  A?   ^^^3        ^^^       ^^^6       ^^y       ^^^^ 

13.  A  certain  number  is  contained  in  i,  f  times;  how 
many  times  is  it  contained  in  2  ? .  in  3  ?  in  5  ?  in  8  ?  in  20  ? 

in  a  ?  in  c  ?  in  f  ?  in  #  ?  in  5^  ?  in  t\  ?  in  ^  ?  in  -  ?  in  -  ? 

4  »  8  lb  J  ^  ^ 

14.  How  many  times  is  f  contained  in  the  following  num- 
bers? 

(i)inf?  (4)  info?  (7)  in  a?  /n^  ;„ -? 

^^tl  [5)in.v?  (3)^^^  ^'^     -y 

(3)mi?  (6)mi2?  ^^       b 

15.  Find  the  result  in  the  following  indicated  divisions: 
(i)     l^f;  (3)    f^f;  /.N    ^^i.  (fy     c^^ 

(2)  f-^A;      (4)   l-i;        ^^^   b'x'       ^""^  ~x']- 

16.  Make  a  rule  for  dividing  one  fraction  by  another. 

17.  Apply  your  rule  to  the  following: 

(i)    l-5-f;  (.     w_^5.  /»^»j,  c^a 

(2)    A-^4;         ^^'^     I   't'  ^^^    q'  n'  ^^^    X-  f 

Principle  II. — Fractions  are  multiplied  by  multiplying 
their  numerators  for  the  numerator  of  the  product,  and  multi- 
plying their  denominators  for  the  denominator  of  the  product. 

Principle  III. — One  fraction  is  divided  by  another  by 
multiplying  the  dividend  by  the  inverted  divisor;  that  is,  by 
the  reciprocal  of  the  divisor. 

Queries. — (i)  Why  is  the  divisor  inverted? 

(2)  Why  is  the  inverted  divisor  multiplied  by  the  dividend  ? 


CHAPTER  VI 

USES  OF  THE  EQUATION 

§  14.     Problems 

1.  Divide  a  pole  20  ft.  long  into  two  parts  so  that  one  part 
shall  be  4  times  as  long  as  the  other. 

Arithmetic  Solution 
The  shorter  part  is  a  certain  length. 
The  longer  part  is  four  times  this  length. 
The  whole  pole  is  then  five  times  as  long  as  the  shorter  part. 
The  pole  is  20  ft.  long. 
The  shorter  part  is  ^  of  20  ft,  or  4  ft. 
The  longer  part  is  4  •  4  ft.,  or  16  ft. 
Hence,  the  parts  are  4  ft.  and  16  ft.  long. 
Algebraic  Solution 
Let  X  be  the  number  of  feet  in  the  shorter  part, 
then  4X  is  the  number  of  feet  in  the  longer  part, 
and  X+4X,  or  55(;=2o, 
x=  4, 
4x=i6. 
Hence,  the  parts  are  4  ft.  and  16  ft.  long. 

2.  A  farmer  wishes  to  enclose  a  rectangular  pen  with  80 
ft.  of  wire  fencing.  He  wishes  it  to  be  three  times  as  long  as 
it  is  wide.     How  long  shall  he  make  each  side  ? 

Algebraic  Method 
Let  X  be  the  number  of  feet  in  the  smaller  side, 
then  ^x  is  the  number  of  feet  in  the  longer  side, 
and  x-\'^x,  ov  4x  is  number  of  feet  half-way  round  the 
pen. 

4^=40, 
:v=io, 
3^=3°- 
Hence,  the  sides  are  10  feet  and  30  feet  long. 

29 


30  First-Year  Mathematics 

3.  James  has  3  times  as  many  cents  as  Charles,  and  4 
times  as  many  as  WilUam.  All  together  they  have  57  cents. 
How  many  cents  has  each  ? 

4.  A  boy  sold  a  certain  number  of  newspapers  on  Monday, 
twice  as  many  on  Tuesday,  10  more  on  Wednesday  than  on 
Monday,  and  24  on  Thursday.  He  sold  94  in  the  four  days. 
How  many  did  he  sell  on  each  day  ? 

5.  A  man  divides  up  his  160  acre  farm  as  follows:  He 
takes  a  certain  number  of  acres  for  lots,  4  times  as  much  for 
pasture,  4  times  as  much  for  corn  as  for  pasture,  \  as  much 
for  wheat  as  for  corn,  and  15  acres  for  meadow.  How  many 
acres  does  he  assign  to  each  purpose  ? 

6.  A  May-pole  22  ft.  high  breaks  into  two  pieces  so  that 
the  top  piece,  hanging  beside  the  lower  piece,  lacks  6  ft.  of 
reaching  the  ground.     How  long  is  each  piece  ? 

7.  A  pony,  a  saddle,  and  a  bridle  together  cost  $120.  The 
bridle  costs  \  as  much  as  the  saddle,  and  the  pony  costs  $12 
less  than  12  times  as  much  as  the  saddle.'  What  was  the  cost 
of  each  ? 

8.  A  bicyclist  rode  a  certain  number  of  miles  on  Monday, 
f  as  many  miles  on  Tuesday,  f  as  many  on  Wednesday,  f  as 
many  on  Thursday,  as  many  on  Friday  as  on  Monday,  and 
20  miles  on  Saturday.  On  the  six  days  he  rode  152  miles. 
How  many  miles  did  he  ride  each  day  ? 

9.  The  area  of  a  triangular  piece  of  ground  is  315  sq.  rd. 
One  side  is  30  rods.  How  long  is  a  fence  at  right  angles  to 
this  side  from  the  opposite  corner  ?     (Use  principle  below.) 

Principle. — The  area  of  a  triangle  is  equal  to  \  the  pro- 
duct 0}  its  base  and  altitude. 

10.  If  this  fence  divides  the  side  (30  rods)  so  that  one  part 
is  twice  as  long  as  the  other,  what  are  the  areas  of  the  two  lots  ? 

11.  If  the  fence  divides  the  side  (30  rods)  so  that  one  part 
is  five  times  the  other,  what  are  the  areas  of  the  two  lots  ? 

12.  A  local  train  goes  at  the  rate  of  30  miles  an  hour.     An 


Uses  of  the  Equation  31 

express  starts  two  hours  later  and  goes  at  the  rate  of  50  miles 
an  hour.  In  how  many  hours,  and  how  far  from  the  starting 
point  will  the  second  train  overtake  the  first  ? 

13.  A  book  dealer  has  in  stock  twice  as  many  Readers 
as  Arithmetics,  four  times  as  many  Readers  as  Histories.  In 
all  he  has  70  Readers,  Arithmetics,  and  Histories.  How  many 
of  each  has  he  ? 

§  15.     The  Sum  of  the  Three  Angles  of  Any  Triangle 

Draw  any  triangle.  From  the  vertex  of  one  angle  draw 
a  perpendicular  to  the  opposite  side.  Cut  out  the  triangle 
and  fold  so  that  the  vertices  all  meet  at  the  foot  of  the  perpen- 
dicular. What  seems  to  be  the  sum  of  the  three  angles  of 
the  triangle  ?    This  is  sometimes  called  Pascal's  Method.    Is 


Fig.  14 

the  sum  of  all  the  angles  formed  about  a  point  on  one  side  of  a 
straight  line  always  equal  to  the  two  right  angles,  or  180  de- 
grees ? .  Fig.  14. 

Draw  any  triangle.  Lay  a  pencil  flat  along  one  side  of 
the  triangle  and  note  which  way  it  points.  Revolve  it  about 
one  vertex  as  a  pivot  across  the  triangle  till  it  coincides  with 
the  next  side,  continue  this  process  at  the  two  other  vertices. 
Through  what  part  of  a  circle  has  the  point  of  the  pencil  re- 
volved ?     Does  this  seem  to  verify  Pascal's  Method? 

I .  Find  the  value  of  each  angle  of  a  triangle  in  right  angles 
or  degrees,  if  the  first  angle  is  twice  the  second,  and  the  third 
is  three  times  the  first. 


32  First-Year  Mathematics 

2.  One  acute  angle  of  a  right  triangle  is  f  of  a  right  angle. 
Find  the  other. 

§  i6.     Angles  Made  by  Two  Intersecting  Straight  Lines 

1.  One  angle  of  two  intersecting  straight  lines  is  f  of  a 
right  angle.     Find  the  other  three. 

Fold  two  intersecting  straight  creases  in  a  paper.  Fold 
through  the  vertex  and  bring  the  upper  side  of  one  vertical 
angle  along  the  upper  side  of  the  other.  How  do  the  vertical 
angles  compare  as  to  size?  Treat  the  other  vertical  angles 
in  the  same  way.  State  a  principle  about  the  relative  size  of 
vertical  angles. 

2.  One  of  the  angles  formed  by  two  intersecting  straight 
lines  is  15  degrees.     Find  the  other  three. 

3.  One  of  the  angles  formed  by  two  straight  lines  is  \  R.  A. 
Find  the  other  three. 

4.  The  sum  of  one  angle  and  its  vertical  angle  is  ten  times 
the  sum  of  the  other  two  angles.   Find  the  size  of  each  angle. 

5.  The  difference  between  one  pair  of  vertical  angles  and 
the  other  is  28°.     What  is  the  value  of  each  angle  ? 

§  17.     The  Sums  of  the  Angles  of  Polygons 

1.  Make  any  quadrilateral  and  divide  it  into  two  triangles 
by  drawing  a  diagonal.  What  is  the  sum  of  all  the  angles  of 
the  quadrilateral  ? 

2.  Make  a  pentagon,  and  draw  all  diagonals  possible  from 
one  vertex.  What  is  the  sum  of  the  angles  of  the  pentagon  ? 
What  is  the  sum  of  the  angles  of  a  hexagon  ?  An  octagon  ? 
An  «-gon  ? 

3.  Into  how  many  triangles  is  any  polygon  divided  by  the 
diagonals  from  a  single  vertex  ? 

4.  State  a  law  as  to  the  number  of  right  angles  in  relation 
to  the  number  of  sides  of  a  polygon  ? 


Uses  of  the  Equation  33 

5.  Make  polygons  of  various  numbers  of  sides.  From  a 
point  within  each  polygon  draw  lines  to  all  of  its  vertices. 

Find  the  sum  of  the  angles  and  deduct  the  number  of  right 
angles  formed  about  the  point  from  which  the  Unes  were  drawn. 
How  many  right  angles  does  that  leave  for  the  angles  of  the 
polygon  ?  State  a  law  regarding  the  sum  of  the  angles  of  any 
polygon. 

Interpret  S  —  {n—2)  times  180°,  as  expressing  the  sum  of 
the  angles  of  any  polygon,  if  n  is  the  number  of  sides. 

6.  Using  your  law,  find  the  sum  of  the  angles  of  a  polygon 
of  16  sides;  of  25  sides;  of  20  sides;  of  39  sides. 

7.  Find  the  number  of  sides  of  the  polygon  in  which 
5=7,200°;  5=40  R.  A.;  5  =  190  R.  A. 

8.  Find  the  value  of  one  angle  of  an  equiangular  polygon 
of  ten  sides. 

9.  How  many  sides  has  an  equiangular  polygon  in  which 
one  angle  is  1 50°  ? 

10.  How  many  sides  has  an  equiangular  polygon  in  which 
the  sum  of  two  angles  is  270°? 

§  18.     The  Exterior  Angles  of  Regular  Polygons, 

An  exterior  angle  of  a  polygon  is  formed  between  one  side 

produced  and  the  adjacent  side.     In  Fig.  15,  CAB,  or  FAD, 

is  the  exterior   angle.     Does  _ 

the  extension   of  one  side  or 

the    other    at    any    vertex, 

make   any   difference   in  the 

size    of    the    exterior  angle  ? 

Why? 

I.  Revolve    a    pencil  „ 

^  Fig.  15 

through  each    exterior  angle 

of  any  polygon.  Through  what  part  of  a  circle  does  the 
pencil  point  revolve  ?  What  do  you  conclude  as  to  the  sum 
of  the  exterior  angles  of  any  polygon  ? 


34  First-Year  Mathematics 

2.  How  many  sides  has  an  «-gon  in  which  the  sum  of  the 
interior  angles  is  seven  times  the  sum  of  the  exterior  angles  ? 

3.  How  many  sides  has  an  equiangular  n-gon  in  which  one 
exterior  angle  is  f  of  a  R.A. 

4.  As  the  sum  of  the  exterior  and  interior  angles  at  each 
vertex  of  an  «-gon  is  2  R.A.,  if  you  deduct  the  exterior  angles 
from  the  sum  of  both  exterior  and  interior  angles  at  all  the 
vertices,  what  is  left  as  the  sum  of  the  interior  angles  of  an 
w-gon  ?     Compare  this  with  your  previous  conclusion  on  p.  33, 


CHAPTER  VII 

USES  OF  INEQUALITIES 

§  19.     Laws  for  Use  of  Expressions  of  Inequality 

The  form  4>3  is  read  "four  is  greater  than  three,"  and 
the  form  3<  5  is  read  "three  is  less  than  five." 

Since  4>3,  then  4  +  2>3  +  2,  and  4  — 1>3  — i.  Or, 
4  +  ^>3+^,  and  4— />3— /. 

Moreover,  if  a>b,  then  a+c>b-{-c,  and  a—c>b—c. 

These  examples  illustrate  the  facts:  (i)  that  if  equal  num- 
bers are  added  to  unequal  numbers,  the  sums  are  unequal  in 
the  same  way  (or  order) ;  and  (2)  if  equals  are  subtracted  from 
unequals  the  remainders  are  unequal  in  the  same  way  (or  order) . 

As8>5,so2  •  8>2  •  5, and  — 2  •  8<  — 2  •  5. 

This  example  illustrates  the  principle  that,  if  unequal  num- 
bers are  multiplied  by  the  same  or  by  equal  numbers,  the 
results  are  unequal  in  the  same  order,  if  the  multiplier  is  posi- 
tive, and  the  results  are  unequal  in  the  opposite  order  if  the 
multiplier  is  negative.  I.e.,  if  cyd  and  a  be  any  positive 
number,  then  (i)  ac>ad  and  (2)  —ac<.  —ad. 

Proof. — If  c>d,  then  c—d  is  a  positive  number.  Why? 
Hence  a{c—d)  is  positive  (a  being  positive),  or  ac—ad  is  posi- 
tive and  acyad.     Why? 

—a(c—d)  is  a  negative  number.  Why?  Hence, 
—ac  —  (—ad)  is  a  negative  number.    Then  —ac<  —ad.    Why  ? 

Since  dividing  is  the  same  as  multiplying  by  the  reciprocal 
of  the  divisor,  the  last  principle  is  true  for  dividing  unequal 
numbers  by  the  same  or  by  unequal  numbers. 

§  20.     Problems  with  Inequalities 

I .  Of  what  different  whole  numbers  is  it  true  that  one-half 
of  the  number  increased  by  5  is  greater  that  four  times  four- 
thirds  of  it  diminished  by  three  ? 

35 


36  First-Year  Mathematics 

2.  A  factory  finishes  a  certain  number  of  wagons  a  day.  If 
6  more  each  day  were  finished  it  would  make  more  than  240 
a  week.  If  16  less  each  day  were  done  there  would  be  an  out- 
put of  less  than  half  of  240  a  week.  How  many  were  finished 
each  day  ? 

3.  If  a>b  and  b>c,  prove  a>c. 

4.  The  sum  of  the  squares  of  any  two  different  numbers 
is  greater  than  twice  their  product. 

Proof.  (a—by>o,  because  the  square  of  any  number 
is  a  positive  number.  Therefore,  a'  —  2ab  +  b'>o.  Adding 
2ab  =  2ab,  the  result  is  a'  +  b'>2ab. 

5.  Prove  that  the  sum  of  the  squares  of  any  number  (ex- 
cept i)  and  its  reciprocal  is  greater  than  2. 

Suggestion : 


c-^y 


>o.     Why? 


6.  Is  any  side  of  a  triangle  less  than  the  sum  of  the  other 
two  ?    Prove.     Greater  than  the  difference  ?    Prove. 

7.  Prove  in   the   accompanying   figure   that    BD-|-DC< 

BA-hAC. 

8.  Prove  the  sum  of  three 
fines  drawn  from  a  point 
within  a  triangle  to  the  three 
vertices  is  less  than  the  peri- 
meter of  the  triangle. 
^'^^-  i^  9.  An  exterior  angle  of  a 

triangle  is  greater  than  either  of   the  interior  non-adjacent 
angles.     Prove. 

10.  Prove  angle  D>  angle  A,  Fig.  16. 


CHAPTER  VIII 

THE  OPERATIONS  OF  ARITHMETIC  ABBREVIATED 

§21.     General  Arithmetic 

1.  Denote  the  minuend  by  w,  the  subtrahend  by  s,  and 
difference  hy  d.  Show  by  an  equation  the  relation  of  these 
numbers. 

Answer:  Equation,  d=m—s.  Interpretation:  "difference 
equals  minuend  minus  subtrahend." 

2.  The  multipUcand  is  M,  the  multipher  m,  and  the  pro- 
duct P.  Express  their  relation  by  an  equation  and  translate 
the  equation  into  words. 

3.  Add  s  to  both  sides  of  the  equation  d=m—s  and  trans- 
late the  result.  Show  that  the  result  is  a  rule  for  checking,  or 
testing,  subtraction. 

Note. — Observe  that  s—s,  or  — 5-h^=o. 

4.  Divide  both  sides  oi  P=M  •  m,  or  f*=Mm,  (a)  by  m 
and  interpret  (translate  into  words)  the  result,  (b)  by  M,  and 
interpret  the  result. 

5.  Show  by  an  equation  the  relation  between  the  dividend, 
D,  divisor,  d,  and  quotient,  q.     Interpret  the  equation. 

6.  Show  by  an  equation  the  relation  of  the  dividend  D, 
divisor  d,  quotient  q,  and  remainder  r,  and  interpret  the  equa- 
tion. 

Express  by  equations  the  relations  of  the  numbers  in  the 
following  problems: 

7.  A  boy  has  m  marbles  and  buys  b  more*  He  then  has 
M  marbles. 

8.  There  are  h  boys  and  g  girls  in  a  class  of  p  pupils. 

9.  A  boy  earns  c  cents  a  day  for  d  days.  He  then  has  C 
cents. 

10.  A  rectangular  flower  bed  is  /  feet  long  and  contains 
s  square  feet.     The  width  is  w  feet. 

37 


38  First-Year  Mathematics 

11.  A  bicyclist  rides  M  miles,  which  is  r  miles  more  than 
/  times  m  miles. 

fi  ft 

12.  The  sum  of  the  fractions  ~  and  — '  is  S. 

di  a, 

i^.  The   difference   of   the   fractions  -^    and    -7-    is    Z) 
^  dj,  d. 


is  the  minuend-fraction 


til  ^a 

14.  The  product  of  the  fractions  -j-  and  -;-  is  P. 

15.  The  base  of  a  rectangle  is  b  ft.  and  the  altitude  is  a  ft. 
The  area  is  R  sq.  ft. 

16.  Each  side  of  a  square  is  5  ft.  and  the  area  is  A  sq.  ft. 

17.  The  value  of  a  load  of  corn  of  b  bu.  at  c  ct.  per  bu.  is 
D  cents;  D  dollars. 

18.  The  quotient  of  the  fraction  -7^  divided  by  -7^  is   Q. 

19.  A  decimal  fraction  has  three  units  in  tenths  place,  and 
5  units  in  hundredths  place,  and  its  value  is  v. 

Answer:  v=T%+Tto- 

20.  A  decimal  fraction  has  /  units  in  tenths  and  h  units 
in  hundredths  place  and  its  value  is  v. 

21.  A  decimal  has  a  units  in  tenths  place  and  b  units  in 
hundredths  place  and  c  units  in  thousandths  place.  Its  value 
is  V. 

22.  A  mixed  number  has  a^  units  in  hundredths  place,  a, 
units  in  tenths  place,  a^  in  units  place,  a^  in  tens  place,  and 
O5  in  hundreds  place  and  05  in  thousands  place.  Its  value 
is  V. 

23.  The  base  is  b  bushels,  the  rate  r  per  cent.,  and  the 
percentage  p  bu. 

24.  The  base  is  b  lb.,  the  rate  r  per  cent.,  and  the  percent- 
age p  lb. 

25.  The  base  is  b,  the  rate  r,  and  the  percentage  p. 


The  Operations  of  Arithmetic  Abbreviated  39 


b-r 

1.  Divide  both  sides  of  the  equation  p= by  b  and 

100 

br 

2.  Multiply  both  sides  of  the  equation  />= by   100, 

100 


§  22.     Laws  of  Percentage  and  Interest 

de  both  s 
interpret  your  result. 

2.  Multiply  both 

then  divide  both  sides  by  r,  and  interpret. 

br 

3.  Multiply  both  sides  of  the  equation  p= by   100, 

then  divide  both  sides  by  b  and  interpret  your  result. 

4.  Express  by  an  equation  the  relations  involved  if  the 
interest  is  %i,  the  principle  %p,  the  rate  r%  and  the  time  t  yrs. 

prt 

5.  Divide  both  sides  of  the  equation  i= by  rt,  and 

then  multiply  by  100,  and  interpret. 

6.  Divide  both  sides  of  the  equ 

multiply  both  sides  by  100,  and  interpret. 

.     Prt 

7.  Multiply  both  sides  of  the  equation  *=^ —  by  100, 

then  divide  by  pt  and  interpret. 

Query. — What  is  the  product  of  any  fraction  by  a  number 
equal  to  its  own  denominator  ? 


prt 

6.  Divide  both  sides  of  the  equation  i= by  pr,   then 

100 


CHAPTER  IX 

THE  EVALUATION  OF  EXPRESSIONS 

§  23.     The  Circle  and  Sphere 

1.  The  circumference    of    a  circle  is   equal  to  ^  of  the 
diameter;  i,  e.,  C='jr  d,  where  ir=^. 

Find  C,  if  (i)  rf  =  2i  ft.; 

(2)  d  =  Tit.; 

(3)  rf=ifoot;       . 
Find  i,  if  (i)  c=88ft.; 

(2)  c  =  66ft.; 

(3)  c  =  i6  feet. 

2.  If  r  stands  for  the  radius  of  a  circle  and  C  for  its  cir- 
cumference, from  problem  i,  show  that  C  =  2  tr  r. 

Find  C,  if  (i)  r=3  ft.; 

(2)  r=6  ft.; 

(3)  r= 18  feet. 

3.  The  area  of  a  circle  equals  \'^  times  the  square  of  the 
radius;  i.  e.,  A=tt  r^,  where  7r=\^. 

Find  ^,  if  (i)    r=3  ft.; 

(2)  r  =  7ft.; 

(3)  r  =  2i  feet. 
Find  r,  if    (i)  ^  =  154  sq.  ft.; 

(2)  A  =220  sq.  ft.; 

(3)  .4  =  86.4  square  feet. 

4.  The  volume  of  a  sphere  equals  f  ir  times  the  cube  of 
the  radius;  i.  e.,  V=^  -k  r^,  where  7r=^. 

Find  F,  if  (i)  r=ift.; 

(2)  r  =  ii  ft.; 

(3)  ^=2  feet. 
40 


The  Evaluations  of  Expressions  41 

§  24.     Motion  and  Mensuration 

1.  The  distance  traversed  by  a  moving  body  is  equal  to 
the  rate  multiplied  by  the  time;  i.e.,   D=rL 

Find  D,  if    (i)  ^=30  ft.  per  second,  and  ^=5  seconds; 
(2)  r=5  mi.  per  hour,  and  t  =  i'j  hours. 

2.  The  area  of  a  rectangle  is  equal  to  the  product  of  the 
base  and  the  altitude;  i.  e.,  A  =ba. 

Find  A,  if    (i)  6  =  13  ft.,  and  a  =  24  ft.; 

(2)  6  =  10.2  in.,  and  a =3. 5  inches. 

3.  The  area  of  a  triangle  is  equal  to  ^  the  product  of  the 
base  by  the  altitude;  i.  e.,  A=^  ba. 

Find  ^,  if    (i)  6  =  12  ft.,  and  a  =  16  ft.; 

(2)  6=8.2  rd.,  and  0  =  7.78  rods. 

4.  The  area  of  a  parallelogram  is  equal  to  the  product  of 
the  base  by  the  altitude;  i.  e.,  A  =ba. 

Find  A,  if    (i)  6  =  28,  and  0  =  19; 

(2)  6  =  16.3,  ^^^  0  =  14.6 

5.  Find  the  numerical  value  of  each  of  the  following  ex- 
pressions, when  a  =  5,  6=3,  c  =  io,  w=4,  n  =  i: 

,    X  /^x     C-I-2W 

(0  scm';  (6)  ;-^; 

(2)     3    a3c;  (y)     a2_|_52_|_^2_|_^2_|_„2. 

(3)  13  6^c«^;  (8)  2ab-\-4bc  +  $cm''; 

(4)  w5  +  i;  (9)  a3-b3  +  c3; 

(5)  5  abc+;^m^n3;  (10)  3  abcmn. 

6.  A  stone,  falling  from  rest,  goes  in  any  given  time  16  ft. 
multiplied  by  the  square  of  the  number  of  seconds  it  has  fallen ; 
i.  e.,  5  =  16  P. 

Find  s,  if    (i)  t=4  seconds; 

(2)  /  =  ii  .5  seconds. 

7.  A  stone  thrown  downward  goes  in  any  given  time 
16  ft.  multipUed  by  the  square  of  the  number  of  seconds  it 


42  First-Year  Mathematics 

has  fallen,  plus  the  product  of  the  velocity  with  which  it  is 
thrown  and  the  number  of  seconds  fallen;  i.  e.,  s  =  it  t'+vt. 

Find  5,  if    (i)  t=i2  seconds,  and  v=;^  ft.  per  second; 
(2)  /=  8  seconds,  and  v='j  ft.  per  second. 

8.  The  time,  /,  taken  for  a  pendulum  to   make  a   single 
vibration  equals  -n-^-  ,  where  /  is  the  length  of  the  pendulum 

o 

in  feet,  ^  is  32,  and  w  is  -^. 

Find/,  if    (i)  /=8ft.; 

(2)  /=f|f  feet. 

Find  /,  if    (i)  /  =  i  second  ; 
(2)  /=4  seconds . 

9.  The  area.  A,  of  a  triangle  is  equal  to 


\/s(s-a)(s-b){s-c) 
where  a,b,c  represent  the  lengths  of  the  sides  and  5 = ^(a + 6 + c) . 
Find  ^,  if  (i)       a=s,     b=4,         c  =  5  ; 
(2)       a  =  5,     &  =  i2,        c  =  i3. 

10.  il  =  -^.   Findfl",  if  (i)     /  =  ii,        /  =  5,     S  =  2; 
5500 

(2)     /  =  io,         1=33  y  S  =  ^' 

'=( 

(2)    r=i3,        /  =  i5 


i?=^.     FindR,if(i)    r=ii.5,    r'  =  6.5 


MV 

12.  E=^^^  .     FindE,  if  (i)ir  =  i2,  F  =  5; 

2 

(2)M  =  ii,  V=g. 

Find  M,  if  (i)    £=8,  F=4  ; 

(2)   £  =  50,  F  =  5- 

13.  V  =  ^h{B+b+VBb). 

Find  F,  if  (i)  /t=9,      B  =  i6,  6=4; 

{2)  h  =  6,      B  =  i6,  b=g. 


The  Evaluations  of  Expressions  43 

14.  Find  the  numerical  value  of  each  of  the  following 
expressions,  when  a=8,  6=6,  c  =  i,  x=h,  y=^: 


CHAPTER  X 
GEOBIETRIC  REPRESENTATION  OF  QUANTITY 
§  25.  Drawing  to  Scale 

1.  How  would  you  draw  a  map  of  a  rectangular  field  50  ft, 
long  and  40  ft.  wide  on  a  piece  of  paper  8  in.  by  10  in.  ?  What 
relation  would  exist  between  the  field  and  the  map  ? 

2.  Draw  a  line  3  inches  long,  and  let  it  represent  a  distance 
of  48  ft.  What  distance  would  be  represented  by  i  inch  ? 
by  2  inches  ?  by  6  inches  ?  by  i^  inches  ?  by  2f  inches  ?  by 
lyV  inches  ? 

Note. — In  the  above  example  the  drawing  is  said  to  be 
made  to  a  scale  0}  i  inch  to  id  feet. 

3.  Draw  to  the  same  scale:  8  ft.;   12  ft.;   24  ft.;  28  feet. 

4.  If  a  line  5  inches  long  is  taken  to  represent  a  distance 
of  75  miles,  what  scale  is  used? 

5.  Make  a  drawing  to  the  scale  of  J  inch  to  i  rod  (i''  = 
I  rd.)  of  a  rectangular  field  16  rods  long  and  12  rods  wide. 

6.  Fig.  17  represents  a  seven-sided  field,  and  Fig.  18  is  a 
scale  drawing  of  Fig.  17.     The  distances  from  O  to  the  several 


'3 


A 


A 


Fig.  17  Fig.  18 

corners  of  the  field  represented  in  Fig.  17  are:  to  A,  678  ft.; 
to  B,  6x2  ft.;  to  C,  683  ft.;  to  D,  738  ft.;  to  E,  698  ft.;  to  F, 
625  ft.;  to  G,  679  ft. 

44 


Geometric  Representation  of  Quantity 


45 


If  the  scale  of  Fig.  i8  is  i"  =  2oo'  what  is  the  length  in 
inches,  to  two  decimal  places,  of  the  lines  oa,  ob,  oc,  od,  oe,  of,  og? 

7.  If,  in  Fig.  18,  ab=4.i2  inches,  be =4. 12  inches, 
cd  =  2.86  inches,  de  =  2.75  inches,  ef  =  5.86  inches,  fg  =  6.i8 
inches,  and  ga  =  5.98  inches,  what  are  the  lengths  in  feet  of 
the  corresponding  sides  AB,  BC,  etc.,  of  the  field  ? 

8.  If  the  area  of  triangle  oab  (Fig.  18)  is  4.94  sq.  in.,  what 
would  be  the  area  of  triangle  OAB  (Fig.  17),  if  the  scale  is 
I  inch  to  200  feet  ?  Answer  the  same  question  using  a  scale 
of  I  inch  to  100  feet. 

9.  Draw  to  a  scale  of  J"  =  10  rd.  a  garden  plot  (from  the 
data  of  Fig.  19).     Use  a  protractor  to 
draw  the  angle  A. 

How  many  rods  long  is  the  side  BC  ? 

How  many  degrees  in  the  angle  B  ? 
In  angle  C  ?  What  is  the  sum  of  the 
three  angles  of  the  triangle  ABC? 
What  do  you  infer  from  this  ?  ^ 

10.  A   railroad   surveyor  wishes   to  ^^ 
measure  across  the  swamp  AB  (Fig.  20).     He  measures  the 

distance  from  the  tree  at  A  to  a 
stone  at  C  and  finds  it  to  be  150  ft. 
The  distance  from  the  tree  at  B  to 
the  stone  is  165  ft.  The  angle* 
at  C  is  85°.  Draw  to  a  convenient 
scale  the  triangle  ABC  and 
determine  from  your  drawing  the 
distance  in  feet  across  the  swamp. 

11.  A  man  wishes  to  measure  the  width,  AC,  of  a  stream 
(Fig.  21)  without  crossing  it.  He  lays  off  a  Une,  BC,  on  one 
side  of  the  river,  and  measures  (with  his  transit)  angles  B  and  C. 

*  Note. — Surveyors  measure  angles  with  an  instrument  called  the 
transit.  For  rough  measurements  the  pupil  can  easily  construct  an  an- 
gle-measurer by  tacking  a  protractor  on  a  board.  A  ruler  with  a  pin 
stuck  in  it  at  each  end  can  be  used  as  a  sight. 


46 


First-  Year  Mathematics 


Draw  a  triangle  carefully  from  the  data  given  in  the  figure,  and 
determine  the  width  of  the  river  by  measuring  AC  with  a  ruler. 


Fig.  21 

12.  A  boy  wishes  to  determine  the  height  HK  (Fig.  22)  of 

a  factory  chimney.     He  places  the  angle-measurer  first  at  B 

and  then  at  A  and  measures  the  angles  x  and  y.    The  angle - 

measurer  lies  on  a  box,  or  tripod,  3^  ft.  from  the  ground.     A 

/f 


Fig.  22 


and  B  are  two  points  in  line  with  the  chimney,  and  at  a  con- 
venient distance  apart.  Show  by  a  drawing  how  the  boy  may 
determine  the  height,  HK,  of  the  chimney,  if  the  ground  is 
level,  and  if  :x;=63°,   y=33^'^,  and  AB  =  5o  feet. 


Geometric  Representation  of  Quantity 


47 


13.  Determine  the  height  of  the  flag-staff  on  your  school 
building  by  making  measurements  similar  to  those  used  above. 
For  this  purpose  construct  an  angle-measurer  according  to 
the  directions  in  the  foot  note  on  page  45. 

14.  Find  the  distance  AB  (Fig.  23),  through  some  building 
near  your  school. 
First  select  some 
point  O  from  which 
you  can  see  both  A 
and  B.  Measure 
angle  x  with  the 
angle-measurer. 
Also  measure  OA 
and  OB  with  a  yard-  < 
stick,  or  surveyor's 
tape,  or  chain. 
Make  a  scale  draw- 
ing of  the  triangle 
AOB,  and  measure  the  sid; 
the  actual  distance  AB  ? 

15.  A  surveyor  wants  to  know  the  distance  MQ  (Fig.  24) 


Fig.  23 
AB  wiih  a  rule''. 


How  long  is 


Fig.  24 

between  two  trees  on  the  opposite  bank  of  a  river.  He  places 
his  transit  first  at  A,  measuring  the  angles  x  and  y;  and  then 
at  K,  measuring  the  angles  w  and  z.     Make  a  scale  drawing 


48 


First-  Year  Mathematics 


and  find  the  distance  MQ,  if  AK  =  5o  rods,  angle  :x;=45°, 
angle  ^'  =  115°,  angle  2=30°,  and  angle  w  =  8o°. 

Hint:  First  draw  triangle  AMK,  and  then  draw  triangle 
AQK. 

16.  A  flag-pole  40  ft.  high  casts,  on  level  ground,  a  shadow 
60  ft.  long.  Draw  the  triangle  to  scale  and  measure  the  angle 
which  shows  the  angle  of  elevation  of  sun  at  the  time. 

17.  Draw  at  least  three  or  four  triangles,  ABC,  for  each 
of  the  sets  of  given  parts  in  the  table  below.  The  symbol  ( ^  ) 
means  angle. 

Compasses  are  needed 
for  triangles  IV  and  VI. 
In  the  completed  trian- 
gles draw  heavy  lines  for 
the  given  sides  and  bro- 
ken, or  dotted,  lines  for 
the  other  sides.  Indicate 
the  given  angles  by  an 
arc.  Thus,  for  triangle 
III: 


No. 

AB 

BC 

CA 

ZA 

ZB 

zc 

I 

1" 

2" 

II 

l" 

30° 

III 

1" 

ir 

60° 

IV 

1" 

ir 

2" 

V 

1" 

15° 

30° 

VI 

2" 

ir 

30° 

VII 

li" 

1" 

60° 

30° 

Fig.  24(1 


19. 


;»/^  18.  Can   a    triangle   be    drawn 

whose  sides  are  3",  2",  and  6"  ? 
2r,  41",  and  7''? 
Two  sides   of  a  triangle   are  4"  and  9"  in  length. 
Between  what  Hmits  must  the  length  of  the  third  side  be  ? 

20.  In  I,  above,  are  all  the  triangles  necessarily  of  the 
same  size  and  shape  ?  In  II  ?    In  III  ?  In  IV  ?  In  V  ?  In  VI  ? 

21.  Can  you  make  triangles  by  omitting  any  one  of  the 
four  given  parts  in  VII  ? 


Geometric  Representation  of  Quantity  49 

§  26.     Summary 

Any  triangle  has  six  parts — 3  sides  and  3  angles.  It  is 
to  be  observed  in  drawing  the  triangles  of  the  above  table, 
that  when  less  than  three  parts  of  a  triangle  are  given  many 
triangles  having  them  can  be  drawn.  These  triangles  are, 
however,  not  necessarily  of  either  the  same  size  or  shape. 

It  is  also  to  be  noted  that  when  three  parts  are  given  (with 
two  exceptions  to  be  considered  later) ,  all  the  triangles,  which 
have  these  three  given  parts,  are  necessarily  of  the  same  shape 
and  size.  The  three  given  parts  are  said  to  determine  the  tri- 
angle. Such  triangles  are  really  the  same  triangle  in  different 
positions.  Any  one  of  the  triangles  may  be  so  placed  upon 
another,  that  the  corresponding  parts  of  the  two  triangles  will 
coincide,  or  fit  exactly.  They  are  called  equal,  or  congruent 
triangles. 

§  27.     Exceptional  Cases 

First  Exception — 

If  the  value  of  angle  B  in  VII  is  omitted,  we  have  an 
example  of  the  first  exception  mentioned  above. 

I.  Point  this  out  in  drawings  of  triangles  having  the  fol- 
lowing parts : 

(i)  AB  =  iiin.,  BC  =  i    in.,  and  /A=6o°; 

(2)  AB=3iin.,  BC=3    in.,  and  ^A  =  6o°; 

(3)  AB=4    in.,  BC  =  2^in.,  and  /A=3o°. 

Second  Exception — 

1 .  Draw  three  or  four  triangles  in  each  of  which  the  angles 
are  respectively  35°,  65°,  and  80°.  Are  all  of  those  triangles 
necessarily  of  the  same  size  ?  Do  they  all  have  the  same 
shape  ? 

2.  Draw  three  or  four  triangles  each  of  which  has  angles 
90°,  25,°  and  65°.     Are  all  of  these  triangles  necessarily  of  the 

a  me  size  and  shape  ? 

R 


S©  First-Year  Mathematics 

Triangles  having  the  same  shape  are  called  similar  tri- 
angles.    Similar  triangles  are  not  necessarily  of  the  same  size. 

§  28.     Triangles  Having  the  Same  Shape  (Similar  Triangles) 

1.  Draw  a  triangle  with  sides  2"  and  3'',  respectively,  and 
an  angle  of  50°  included  between  them.  First  draw  it  actual 
size  and  then  to  the  scale  of  \"  =  i".  Measure  with  a  pro- 
tractor the  pairs  of  corresponding  angles.  Are  the  two  tri- 
angles similar  ?    Why  ? 

2.  These  triangles  though  different  in  size  have  the  same 
shape  and  have  their  corresponding  angles  equal.  Notice  also 
that  each  side  of  the  smaller  triangle  is  f  of  the  length  of  the 
corresponding  side  of  the  larger  triangle.  All  similar  triangles 
may  be  regarded  as  the  same  triangle  drawn  to  different  scales. 

They  may  be  regarded  as  the  same  triangle  magnified,  or 
minified  to  a  definite  scale. 

3.  Draw  any  triangle,  and  then  draw  another  one  similar 
to  it,  to  a  scale  of  2  :  i .     Draw  another  to  a  scale  of  ^  :  i . 

4.  Two  rectangular  flower-beds  have  the  same  shape,  but 
are  different  in  size.  One  is  3  ft.  wide  and  5  ft.  long;  the 
other  is  12  ft.  wide.     How  long  is  it? 

5.  Two  books  have  the  same  shape.  One  is  5^  inches 
wide  and  7^  inches  long.  The  other  is  15  inches  long.  How 
wide  is  it  ? 

6.  The  top  of  a  desk  and  a  rectangular  sheet  of  paper  12 
in.  X  18  in.  have  the  same  shape.  The  desk  is  2  ft.  wide. 
How  long  is  it  ? 

7.  A  city  block  and  a  lot  within  the  block  have  the  same 
shape.  The  lot  is  100  ft.  by  150  ft.  and  the  block  is  300  ft. 
wide.     How  long  is  it  ? 

8.  The  gables  of  a  house  and  of  a  porch  have  the  same 
shape.  The  sides  of  the  porch-gable  are  7  ft.,  7  ft.  and  10  feet. 
The  longest  side  of  the  house-gable  is  25  ft.  How  long  are 
the"  other  two  sides  of  the  house-gable  ? 


Geometric  Representation  of  Quantity 


SI 


9.  In  two  triangles  of  the  same  shape,  like  those  of  Fig.  25 
(i),  if  a=4  inches,  A  — 12 
inches  and  6=4  inches,  how 
long  is  5  ? 

10.  In  triangles  of  the 
same  shape,  like  those  of 
Fig.  25  (2),  if  a =4  in., 
A=i2  in.,  and  6  =  5  in., 
how  long  is  5?  If  a=2i 
in.,  h=%  in.,  and  B  =  7,2 
in.,  how  long  \s  A}  If 
a=x  in.,  5=8  in.,  and 
B=2)2  in.,  how  long  is  yl  ? 

11.  In  triangles  of  the 
same  shape,  Uke  those  in 
Fig.  25  (3),  if  ^=21  in., 
6=9  in.,  and  B  =  2'j  in., 
how  long  is  a?  If  a  =  5^ 
in.,  ^=22  in.,  and  -8=30 
in.,  how  long  is  &  ? 

12.  In  triangles  of  the 
same  shape,  like  those  in 
Fig-  25  (4),  if  a=3  in., 
A  =S  in.,  and  C=$  in.,  how 
long  is  c?  If  ^=24  in., 
c=4  in.,  C=7  in.,  how  long 
is  a?  If  ^=;y  in.,  c=4in., 
and  C  =  7  in.,  how  long  is  a  ? 

13.  The  shortest  side  of 
a  triangle  is  12  ft.,  the 
longest  is  24  ft.,  and  the 
third  side  is  16  feet.  If  a 
similar  triangle  has  its  shortest  side  8  ft.,  what  are  the 
lengths  of  the  other  two  sides  ? 


52 


First-Year  Mathematics 


l^^ 


In  Fig.  26,  if  the  stake  3  ft.  high  casts  a  shadow  8  ft. 

long,  and  the  tree, 

at  the  same  time, 

casts  a  shadow  80 

13     ^**^     ft.  long,  how  high 

is  the  tree  ? 

Fig.  26  i^.  Measure 

the  height  of  some  tree,  building,  or  flag-pole,  by  the  method 

of  shadows  as  in  problem  14. 

1 6.  How  long  is  x  in  Fig.  27?   Are  triangles  O  AB  and  OHK 
similar  ?    Give  reason  for 
answer. 

17.  A  boy  holds  a 
pencil,  AB  (Fig.  28),  2^  ft. 
from  his  eye,  so  that  it 
covers  a  flag-pole  360  ft. 
away.  To  make  the  tri- 
angles EAB  and  EFK 
similar,  how  must  the  pencil  be  held  ? 
long,  how  high  is  the  pole  ? 


Fig.  27 

If  the  pencil  is  7  in. 


«^'  £, 


Fig.  28 


18.  A  lumberman  who  is  5  ft.  tall  wishes  to  find  a  tree 
60  ft.  to  the  first  Umbs.  He  drives  a  stake  in  the  ground  and 
places  his  feet  against  it  as  in  Fig.  29.  If  the  stake  is  4  ft. 
high,  how  far  must  it  be  placed  from  the  foot  of  the  tree, 
that  he  may  determine  whether  or  not  the  trunk  is  60  ft.  to 
the  limbs? 

19.  In  Fig.  30,  the  letters  a  and  b  denote  the  same  numbers 


Geometric  Representation  of  Quantity 


S3 


throughout.     How  do  the  areas  of  triangles  I,  II,  and  III 

compare?     I   and 

IV?     II    and    VI? 

Ill  and  V?    Ill  and 

VI  ?    IV  and  VIII  ? 

I    and   VIII?     Ill 

and  IX?      IX  and 

X?     VII    and    X? 

Ill  and  X? 

20.  If  the  altitude 
and  base  of  a  tri- 
angle are  4'  and  15' 


Fig.  29 


54  First-Year  Mathematics 

respectively,  what  are  the  dimensions  of  other  triangles  of  \ 
the  area  ?  of  ^  the  area  ?  of  ^  the  area  ? 

21.  If  the  altitude  and  base  of  a  rectangle  are  6"  and  8" 
respectively,  what  are  the  altitude  and  base  of  triangles  whose 
areas  are  (i)  equal  to  the  area  of  the  rectangle  ?  (2)  twice  the 
area  of  the  rectangle  ?  (3)  \  the  area  of  the  rectangle  ? 

22.  Answer  the  same  questions  when  the  altitude  and  base 
of  the  rectangle  are  a  and  h  inches,  respectively. 

23.  A  triangle  and  a  rectangle  have  equal  bases  and  are 
equal  in  area.    How  do  their  altitudes  compare  ? 

24.  If  the  dimensions  of  a  rectangular  parallelopiped  are 
4' X 1 5^X25',  what  are  the  dimensions  of  other  rectangular 
parallelopipeds  having  \  the  volume  ?  ^  the  volume  ?  \  the 
volimie  ?    ^  the  volume  ?    -^  the  volume  ? 

25.  How  would  you  obtain  the  area  of  the  field  polygon 
in  Fig.  17  ? 


CHAPTER  XI 


W^ 


D 


LUL  I 


EQUATION  APPLIED  TO  SIMPLE  PROBLEMS  ON  BEAMS 
§  29.     Common  Uses  of  Forces 

In  this  chapter  some  practical  problems  arising  out  of  the 
common  uses  of  forces  will  be  solved  by  means  of  the  equation. 
It  is  necessary  first  to  discover  a  law  of  these  forces. 

Arrange  an  apparatus  like  that  in  Fig.  31,  consisting  of  a 
light  wooden  bar  10''  long,  provided  with  pegs  (small  nails). 
A  cord  passes  over  the  pulley,  K,  and  is  attached  at  one  end 
to  the  middle,  M,  of  ] 
the  bar,  and  at  the 
other,  to  a  light  scale- 
pan,  S.  Sufficient 
weight  is  placed  in 
the  pan,  S,  to  hold 
the  bar  in  balance. 
If,  now,  weights  are 
attached  to  the  various 
pegs  on  the  bar,  and 
other  weights  are 
attached  to  the  bottom  Fig.  31 

of  the  pan,  S,  the  bar  will  be  acted  upon  by  forces  of  two 
kinds.  The  weights  on  the  pegs  will  pull  downward  on 
the  bar,  while  those  attached  to  the  pan  will  pull  upward 
on  the  bar  by  the  aid  of  the  cord  and  pulley.  To  avoid  con- 
fusion, an  upward  pulhng  force  will  be  called  positive  and  a 
plus  (+)  sign  will  be  written  before  the  number  showing  its 
strength.  On  the  other  hand,  forces  pulling  downward  on 
the  bar  will  be  called  negative  and  a  minus  (— )  sign  will  be 
written  before  the  magnitude  number. 

55 


M 


f 


56  First-Year  Mathematics 

§  30.     Experiments 

1.  Putting  a  weight,  x,  at  I3,  and  an  equal  weight,  x,  at 
Tj,  it  will  be  found  that  two  weights,  each  equal  to  x,  hung  to 
the  pan,  S,  will  hold  the  apparatus  in  balance.  Beginning 
on  the  left,  record  the  relations  for  balance  thus: 

(i)   —x-\-2x—x=o. 

2.  With  two  weights,  each  equal  to  :v  at  I3,  and  two  weights, 
each  equal  to  x,  at  rj,  4  weights,  each  equal  to  x,  must  be  put 
at  S  for  balance.  But  3  or  5  weights,  x,  at  S  will  be  found 
not  to  balance.     Make  the  record  and  interpretation  thus: 

Record  Interpretation 

(i)   — 2X+4:x:— 2:x;=o  Balance 

(2)  —2X-\-^x—2X=—x       Movement  downward 

(3)  —2x-\-$x—2X=-j-x       Movement  upward. 

3.  If  2  weights,  :x;,  be  put  at  I3,  i  weight,  x,  at  1^,  i  weight, 
X,  at  ta,  and  2  weights,  x,  at  rj,  6  weights,  each  equal  to  x,  at 
S  will  balance  the  apparatus,  but  neither  5  nor  7  weights,  x, 
will  balance  it.     The  results  will  run  thus: 

Record  Interpretation 

(i)   —2x—x-{-6x—x-~2x=o  Balance 

(2)  —2x—x+$x—2x—x=—x  Bar  moves  downward 

(3)  —.2x—x+'jx—x—2x=+x  Bar  moves  upward. 

4.  If  3  weights,  X,  are  placed  at  U  and  2  weights,  x,  at  r,, 
it  will  be  found  that  5  weights,  x,  at  S  will  give  balance  while 
neither  4  nor  3  weights,  x,  at  S  will  do  so. 

Record  as  above. 

5.  With  2x  at  li,  ^x  at  U,  2x  at  r3,  and  2X  at  r^,  the  fol- 
lowing weights  were  tried  successively  at  S:  jx;  8x;  ii^c;  lorv,  gx. 

Write  the  equation  for  each  case  and  interpret  it. 

6.  Record  the  equations  for  these  loadings:  x  at  I4,  2X 
at  I3,  3^1;  at  la,  4X  at  Tj,  4X  at  r3,  and  the  following  weights, 
in  turn,  on  the  pan:  lorv;  125c;  i^x;  i43£;;  i6x.  Interpret 
the  equation  in  each  case, 

7.  Write  down   the   appropriate  equations  and  state  the 


Equation  Applied  to  Simple  Problems  on  Beams        57 


results  as  shown  by  a  beam  for  each  loading  of  the  following 
table: 


No. 

14 

I3 

1. 

ii 

s 

ri 

Ta 

r3 

U 

I 

X 

X 

X 

X 

8^ 

X 

X 

X 

X 

II 

X 

X 

0 

0 

3^ 

0 

0 

X 

X 

III 

y 

0 

0 

0 

2y 

0 

0 

0 

y 

IV 

y 

0 

0 

y 

6y 

y 

0 

0 

y 

V 

2y 

0 

sy 

0 

^sy 

sy 

0 

2y 

0 

VI 

2y 

y 

ly 

y 

i8y 

gy 

0 

sy 

0 

VII 

0 

X 

y 

0 

2X+2y 

0 

X 

y 

0 

VIII 

3« 

a 

3« 

0 

25a 

iSa 

0 

a 

0 

8.  Use  the  same  bar,  supported  as  shown,  and  balance  it. 
If    now   a   weight,    x,  is   placed   at   both    L   and   R,    two 

A S 


S 


h     L     f-^     I.  M     Y.       Y.     Y, 


1^  -  ♦  - 


rt\ 


1 


^ 


Fig.  32 


S8 


First-  Year  Mathematics 


weights,  each  equal  to  Ji;  at  M  balance  the  bar.    The  results 
are: 

Record  Interpretation 

-\-x—2x-\-x=o  Balance. 

Try  the  same  weights  at  L  and  R  and  t^x  diX  yL\  x  dX  M. 

9.  Eight  weights,  x,  at  1^,  3  weights,  x,  at  R,  and  5  at  L 
will  balance  the  bar.  If  the  8  weights,  x,  are  at  r^,  then  3 
weights,  X,  at  L  and  5  weights,  x,  at  R  are  needed  for  balance. 
The  results  are : 


Interpretation 
? 

? 


Record 

(i)    ■\-<^X—?)X-\-T,X=0 

(2)   +3:x;— 8x+5.'C=o. 

10.  Write  out  the  equations  and  state  whether  or  not  there 
is  balance  for  these  experiments: 


No. 

L 

14 

13 

U 

li 

M 

Ti 

Ti 

^3 

u 

R 

I 

65c; 

0 

0 

Sx 

0 

0 

0 

0 

0 

0 

231!; 

II 

8x 

0 

0 

Sx 

0 

4X 

0 

0 

0 

0 

4X 

III 

8:x; 

X 

0 

Sx 

0 

4X 

0 

0 

0 

X 

5x 

.IV 

gx 

X 

0 

Sx 

0 

4X 

0 

0 

0 

X 

5^ 

V 

yx 

0 

8x 

0 

0 

0 

0 

0 

0 

0 

X 

VI 

X 

0 

0 

0 

0 

0 

0 

0 

Sx 

0 

yx 

VII 

2X 

X 

X 

0 

0 

0 

0 

0 

X 

X 

gx 

VIII 

SX 

X 

X 

0 

0 

2X 

0 

0 

X 

X 

gx 

IX 

^X 

X 

X 

0 

0 

2X 

0 

0 

X 

X 

lorv 

When  a  bar  is  supported  in  two  places  (A  and  B,  Fig.32), 
it  is  called  a  beam. 


Equation  Applied  lo  Simple  Problems  on  Beams        59 

In  all  these  problems  what  is  the  test  as  to  whether  the 
bar,  or  beam,  balances  ? 

It  will  be  readily  seen  that  in  each  case  of  balance  with 
either  the  beam,  or  the  bar,  the  sum  of  all  the  forces  pulling 
upward  must  be  equal  to  the  sum  of  all  the  forces  pulling  down- 
ward. This  is  the  same  as  saying  that  the  sum  of  all  the  posi- 
tive forces  is  equal  to  the  sum  of  all  the  negative  forces,  for  balance. 

A  sum  such  as  5:^— 8:^+35;,  which  is  made  up,  partly  of 
positive  and  partly  of  negative  numbers,  is  called  an  algebraic 
sum.  The  left  sides  of  all  the  equations  we  have  been  using, 
in  which  it  is  required  to  combine  positive  and  negative 
numbers  into  a  single  sum,  are  examples  of  algebraic  sums. 

Observe  that  in  all  these  problems  the  forces  are  parallel 
to  each  other. 

§31.     First  Law  of  Parallel  Forces 

We  may  now  state  the  first  law  of  force. 

Law  of  Parallel  Forces:  The  algebraic  sum  of  all  the 
forces  acting  upon  the  material  object  {bar,  or  beam)  must  equal 
zero,  for  balance. 

We  shall  refer  to  this  law  as  Law  I  of  Forces. 

§  32.     Practical  Problems 

I.  A  basket  weighing  56  lb.  (Fig.  33)  is  carried  by  two 
boys  who  lift  at  the  ends  of  a  light  stick,  AB,  the  basket  being 
borne  at  the  middle  point,  M.    ;^  ^  -o 

How  much  does  each  boy  lift  ?     'p  j  A 

Solution. — Notice  that     '  '*'  ' 

the  stick  is  in  balance  under  *  **    *  "*" 

the    action  of  three   parallel  ^^°'  33 

forces,  one  pulling  downward  and  two  equal  forces  pulling 
upward.  Letting  x  denote  the  number  of  pounds  lifted  by  each 
boy,  and  using  Law  I  of  Forces,  the  equation  for  balance  is 

+:x;— 56+^=  o  ; 
or,  by  collecting  terms  2:^—56=  o. 


6o  First-Year  Mathematics 

Now  add  56  to  both  sides  of  the  equation,  thus, — 

2^:— 56=  o 
56  =  56 


and  we  have  2«;=56  ; 

whence,  a; =28. 

Result:  Each  boy  lifts  28  pounds. 

2.  Two  boys,  one  lifting  at  each  end,  carry  a  ten-foot  log, 
weighing  12  lb.  per  foot  of  length.    How  much  must  each  boy 

A  M  .5  lift? 


I  -r        Solution. — Notice  that 

V  '    the  entire  weight  of  the  log 

**  '^^^  *  *  may   be  considered  as  con- 

FiG.  34  centrated  at  the  middle.     We 

have  then  what  amounts  to  a  light  bar,  balanced  under  three 

parallel  forces  and  we  write : 

-\-X  —  \20-\-X—      O, 

or  2a;— 120=     o. 

Adding  120  to  both  sides,  2:x;=i2o  ; 

whence,  x=  60 . 

Result:  Each  boy  must  Uft  60  pounds. 

3.  Two  men,  one  lifting  at  each  end  of  an  iron  bar  weighing 
20  lb.,  carry  a  load  of  150  lb.  at  the  middle  of  the  bar.  How 
many  pounds  does  each  man  carry  ? 

Note. — The  bar  being  supposed  to  have  all  its  weight  con- 
centrated at  its  middle  point,  we  may  think  of  the  bar  as  being 
in  balance  under  the  action  of  four  parallel  forces.  Draw  a 
sketch  and  indicate  by  arrows  where  and  how  the  forces  act. 

The  equation  is  then:  -t-  «;— 20— 150+^=  o, 
or  combining  4- 2:^—170=     o  ; 

adding  170  to  both  sides,  2:^=170, 

whence  x=  85 . 

Result:  Each  man  lifts  85  pounds. 


Equation  Applied  to  Simple  Problems  on  Beams        6i 


4.  By  the  aid  of  a  set  of  whiffle- trees  (Fig.  35)  two  horses 
draw  a  Idad  of  650  pounds. 
Show  that  if  the  force  drawing 
backward  (to  the  left)  on  the 
double- tree  is  negative,  and  the 
forces  drawing  forward  are 
positive,  we  may  find  the  pull 
of  each  of  the  four  traces  by 
this  equation: 

-1- 2:^—650 -|- 2:x;=o. 


.UD 


Fig.  35 


Find  the  number  of  pounds,  x,  of  pull  on  each  trace. 

5.  Two  boys  sUp  a  stick  under  one  end  of  a  160  lb.  log, 
and  a  man  Ufts  at  the  other  end  of  the  log.  If  the  boys'  end 
of  the  log  is  on  the  middle  of  the  stick,  for  balance  how  much 
does  the  man  lift?  How  much  does  each  of  the  boys  lift? 
Show  that  the  equation  is:  -|-ic-|-:v— i6o+2:x;=o.  What  does 
X  stand  for  in  the  equation? 

6.  With    two    hand-spikes  An  ff^ 

(AB  and  CD)  four  men,  lifting  i7  H 

equally  at  A,  B,  C,  and   D,  / "- '^ — • 

respectively,  carry  a  log  weigh-  (^ 

ing  600  pounds.  Find  by  means 

of  the    equation    the    amount 

lifted  by  each  man.  "^  „       , 

■'  Fig.  36 


I 


D 


§  33.  Turning-Tendencies  (Leverages) 
A  light  bar  (Fig.  37)  suppUed  with  equally  spaced  pegs, 
is  balanced  about  its  middle  point,  M.  With  a  number  of 
equal  weights,  y,  the  following  experiments  are  performed. 
I.  Hang  a  weight  of  2;y  on  the  peg  li.  This  weight  tends 
to  turn  the  left  end  of  the  bar  downward.  How  much  weight 
must  be  attached  to  the  hook,  H,  to  balance  this  turning-tend- 
ency ?    Now  hang  the  same  weight,  2y  on  peg  1,,  and  meas- 


62 


Fir  St- Year  Mathematics 


ure  its  downward  turning-tendency  by  attaching  to  the  hook, 
H,  a  weight  sufficient  to  balance  the  bar. 

2.  In  a  similar  manner  find  the  downward  turning- tend- 
ency caused  by  the  weight  2y  on  the  peg  I3;  on  1^;  on  Ij. 

3.  Using  a   weight  of  3^,  find  its   downward    turning- 
tendency  when  placed  on  peg  Ij;  on  I3;  on  I3;  on  I4;  on  I5. 

4.  Perform    Experi- 
ment 3,  using  a  weight 

From  Experiments 
I,  2,  3,  and  4,  it  is  clear 
that  when  any  weight 
(as  2y)  is  hung  on  peg 
Ij,  the  turning-tendency 
caused  by  this  weight 
(2y)  is  five  times  as 
I  great  as  when  the  same 
weight  (2y)  is  hung  on 
peg  li;  on  1^  its  turn- 
ing-tendency is  four 
times  as  great  as  on  li ; 
on  I3  it  is  three  times, 
and  on  Ij  it  is  two  times 
as  great  as  on  Ij. 


Fig.  37 


The  same  facts  hold  when  ;^y,  4y,  or  any  other  weight  is 
used.  In  other  words,  with  an  apparatus  like  that  in  Fig.  37, 
the  turning-tendency,  or  leverage,  caused  by  any  weight  is 
measured  by  the  product  of  that  weight  and  the  distance  from 
the  turning  point,  M,  to  the  peg  where  the  weight  hangs. 

5.  What  is  the  turning-tendency  caused  by  a  weight  of 
•jy  hung  on  peg  Ij  ?  on  I3  ?  a  weight  of  gy  on  I4  ?   on  li  ? 

6.  If  you  had  a  bar  long  enough  to  have  12  pegs  on  each 
side  of  M,  what  would  be  the  turning-tendency  caused  by  a 
weight  S)*  on  peg  li 2  ?   'jyon]^?   yonlji? 


Equation  Applied  to  Simple  Problems  on  Beams        63 

7.  Attach  the  cord,  C  (Fig  37),  to  the  peg  r^  and  perform 
experiments  1-4  on  the  right  side  of  the  bar. 

In  experiment  7,  it  is  seen  that  the  same  facts  are  true  on 
the  right  side  as  on  the  left,  but  the  bar  turns,  or  tends  to 
turn,  in  the  opposite  direction.  To  avoid  confusion,  some 
simple  method  of  distinguishing  between  these  two  directions 
of  turning  is  desired. 

Suppose  a  watch  laid  upon  the  page  of  the  book  with  its 
face  up.  When  the  bar  turns,  or  tends  to  turn  with  the  hands 
of  the  watch,  the  turning- tendency  will  be  called  negative 
and  designated— ;  if  it  turns,  or  tends  to  turn,  against  (opposite 
to)  the  watch-hands,  the  turning-tendency  will  be  called 
positive  and  designated  4-. 

In  all  the  experiments  thus  far  performed  the  weights 
which  caused  the  bar  to  turn  were  downward-pulling  weights, 
or  forces.  But  it  is  evident  that  by  arranging  an  apparatus 
as  in  Fig.  38,  below,  forces  can  also  be  made  to  pull  up- 
ward. As  before  (see  p.  55)  we  shall  designate  downward 
puUing  weights  or  forces  by  — ,  and  upward  pulling  forces 
by  -}-. 

The  distance  from  the  turning-point,  M,  to  the  peg  where 
the  weight,  or  force,  acts  will  be  called  the  lever-arm,  or  arm, 
simply,  of  the  force.  Lever-arms  measured  from  the  turning- 
point  toward  the  right  will  be  marked  -f- ;  those  toward  the 
left,  -. 

For  example,  if  the  distance  from  M  to  peg  r^  be  repre- 
sented by  -f-rv  inches,  then  the  distance  from  M  to  the  peg 
r^  will  be  represented  by  -\-^x  inches;  from  M  to  I3,  by  —  3:^ 
inches,  and  so  on. 

If  a  force  of  —  2^  acts  on  peg  I3  its  turning- tendency,  or 
leverage,  would  be  the  product  of  —2y  and  —2,x,  or  6xy.  Since 
the  bar  tends  to  turn  against  the  watch  hands,  the  turning- 
tendency  is  written  -\-6xy.  If  a  force  of  —  2y  acts  on  r^,  its 
turning-tendency  would  be  the  product  of  —2y  and   -|-3;x;. 


64 


First-Year  Mathematics 


or  —6xy,  since  the  bar  in  this  case  tends  to  turn  with  the 
watch-hands. 

8.  In  the  following  twelve  experiments  write  out  the  lever- 
ages, or  turning-tendencies,  for  the  forces  and  arms  indicated. 


Forces 

Arms 

Leverages 

Forces 

Arms 

Leverages 

I 

-2,y 

—  2X 

VII 

-   6y 

-f      X 

II 

-Ay 

-\-2X 

VIII 

-Sy 

—       X 

III 

-iy 

-SX 

IX 

-sy 

+    IX 

IV 

-6y 

-4X 

X 

-Ay 

—  bx 

V 

-  y 

+  SX 

XI 

-ly 

—    2X 

VI 

-8y 

+9X 

XII 

-6y 

—  I2X 

9.  If  loadings  I  and  VII  in  the  above  tables  were  on  the 

apparatus  at  the  same 

time,    would    the   bar 

balance  or  turn  ?    If  it 

turns,  in  what  direction 

would  it  turn  ?   Answer 

similar  questions  for  II 

and  VIII;   V  and  IX; 

,  VI  and  XII. 

,    s^     '         By  the  apparatus  in 

Fig.  38   forces  can  be 

made   to   pull  upward 

on  either  side  of   the 

bar.    If  a  force  of  2^ 

pulls  upward  on  peg  r^^ 

3  its  turning-tendency  is 

the  product  of    ■\-2y  and  +4X,  or  %xy,  and  since  the  bar 

tends  to  turn  against  the  watch-hands,  the  turning-tendency 


Equation  Applied  to  Simple  Problems  on  Beams        65 

is  written  +8:^3'.  On  peg  I4  the  turning- tendency  would  be 
the  product  oi  +2y  and  —4^,  or  —2>xy,  since  in  this  case  the 
bar  would  tend  to  turn  with  the  watch-hands. 

10.  In  the  following  eight  experiments  write  out  the  lever- 
ages for  the  indicated  arms  and  forces: 


Forces 

Arms 

Leverages 

Forces 

Arms 

Leverages 

I 

+  37 

—  2X 

V 

+6y 

-f      X 

II 

+iy 

-ioc 

Yi 

+2>y 

+  7x 

III 

+  y 

+3^ 

VII 

+3>' 

—     2X 

IV 

+  8y 

+9^ 

VIII 

+  6>' 

—  123!; 

1 1 .  If  III  and  VII  in  the  above  tables  are  on  the  apparatus 
at  the  same  time,  will  the  bar  balance  or  turn?  If  it  turns 
in  which  direction  will  it  turn?  Answer  similar  questions 
for  I  and  V;  II  and  VII;  IV  and  VII. 

In  problem  11,  if  the  turning- tendencies  for  I  and  V  are 
added  the  total  turning-tendency  is  —6xy+6xy=o,  which 
says  in  mathematical  language  that  the  bar  does  not  turn. 
If  the  turning-tendencies  for  II  and  VII  are  added,  the 
total  turning-tendency  will  be  +^xy—6xy——ixy,  which 
shows  that  the  bar  does  not  balance,  but  turns  in  the  negative 
direction.  If  two  or  more  forces  are  acting  on  the  bar  at  the 
same  time,  the  total  turning-tendency  is  found  by  adding  the 
separate  turning-tendencies.  If  the  algebraic  sum  is  o  the  bar 
balances.  If  the  sum  is  not  o  the  bar  turns  in  the  direction 
indicated  by  the  sign  of  the  sum. 

12.  In  each  of  the  following  loadings  find  the  total  turning- 
tendency  and  interpret  it:  -1-3^  at  1;  and  —  33/  at  Ij;  +2y 
at  I3  and  — 3  j  at  I2;  +2y  at  r^  and  —2y  at  r^;  +4y  at  Ij  and 
—  ya.t\;  -I-43' at  rj  and  —  3'at  r4;  -|-2;y  at  I3,  — 3^  at  rj,  and 
— 123'  at  Ij;   —  2y  at  r^,  +2y  a.t  I2,  and  -1-53'  at  r^. 


66  First-Year  Mathematics 

§  34.     Second  Law  of  Parallel  Forces 

From  the  experiments  and  problems  just  given  the  second 
important  law  of  forces  becomes  clear;  viz.: 

Law  of  Turning-Tendencies  or  Leverages. — For  bal- 
ance, the  algebraic  sum  of  all  the  turning-tendencies  must  equal 
zero. 

We  shall  refer  to  this  as  Law  II  of  Forces. 

This  law  is  general  and  apphes  to  heavy  bodies  as  well  as 
to  the  light  bar  used  in  the  experiments. 

By  the  aid  of  the  two  laws  of  force,  now  derived,  we  may 
solve  a  variety  of  practical  problems. 

§35.     Practical  Problems 

1.  A  basket  weighing  84  lb.  hangs  on  a  stick  6  ft.  long,  at 
a  point  2  ft.  from  the  middle,  while  it  is  being  carried  by  two 

, ^'... _^    boys,  one  at  each  end.     How  much 

Af does  each  boy  lift  ? 


T^..2-.^  T         Solution. — Let  x  denote   the 

y  I    weight   carried   by   the   boy  at  the 

^*     *  *!!  short  end  and   y  denote  that  borne 

Fig-  39  by  the  other  boy.     Suppose   M  to 

denote  the  middle  point  of  the  stick.  Then,  calling  lever- 
arms  measured  from  M  toward  the  right,  positive;  and  toward 
the  left,  negative,  by  the  first  law  of  forces 

-|-:x;+;y  — 84=0. 
By  the  second  law  of  forces 

(+^)(-3)  +  (-84)(-2)  +  (  +  >')(+3)=         o. 
SimpUfying  these  equations  we  have  {1)  x-{-y=       84; 
and  (2)  —2,x-\-T^y=—i(i2>. 

These  equations  may  be  solved  thus: 
Multiply  both  sides  of  the 

first  by  3  and  get  2,x-\-T^y~     252. 


Equation  Applied  to  Simple  Problems  on  Beams        67 

Add  this  equation  to  equa- 
tion (2),  thus:  —2x+^y=—i6S 
3X+3y=     252 
6y=+  84. 
Therefore                      (3)  ^=-1-  14. 
But                               (i)           x+y=S4; 

or,  by  subtracting  equation   (3)  from  equation  (i),  member 
from  member—  ae=84  — 14  ; 

that  is,  x='jo . 

Therefore  the  boy  at  the  short  end  carried  70  and  the  other 
14  pounds. 

Check:  (i)         70+14=       84, 

(2)   —210+42  =  — 168. 

2.  Solve  a  problem  like  problem  i,  supposing  the  basket 
to  weigh  60  lb.  and  to  hang  at  a  point  2  ft.  from  the  left  end  of 
the  stick;  4  ft.  from  the  left 
end ;  5  ft.  from  the  left  end. 

3.  Two  men  lifting  at  the     ^J         r~~~A1  T 
ends  of  a  stick  8  ft.  long  raise      I         v  I 
a   certain   weight.     What    is  +7^'      ~                                 ♦if 
the  weight,  and  at  what  point                        Fig.  40 

does  it  hang,  if  one  man  lifts  25  lb.,  and  the  other  75  pounds? 
Solution. — The  equations  are : 

From  Law  I,  +25  —  1^+75=0,  or  1^=100. 
Let  X  be  the  number  of  feet  to  the  left  of  the  mid-point  of  the 
bar. 

From  Law  II, 

(+75)(-4)+(-W^)(-^)  +  (  +  25)(+4)=o, 
or,  Wx=  +  2oo , 

I00;»=+200, 
X=+2  . 

The  required  weight  is  100  lb.  and  it  hangs  2  ft.  from  the 


<-- ^- 


68 


First-Year  Mathematics 


1 


-#-x 


middle  toward  the  one  who  Ufts  75  pounds.     Check:    +25 
—  100  +  75=0. 

4.  Suppose  a  bar  10  ft.  long,  weighing  30  lb.,  is  used  by 
two  men,  one  grasping  it  at  each  end,  to  carry  a  load  of  170 

^ ^  ^  /q'  ^ ^  lb.     How  many  pounds 

^~^"\  ^  1  must  each  man  carry,  if 

the   load   is  attached  2 

ft.  from  the  left  end  ? 

-'^^  *^^  V       Note.— The  weight 

Fig.  41  of  the    bar  itself   may 

be  treated  as  a  load  of  30  lb.  hanging  to  the  bar  at  its  middle 

point. 

Suggestion. — Counting  lever  arms  from  the  middle  point 
of  the  bar  the  equations  are: 

(i)  +  X— 170— 30+  y=o, 
(2)   -5^  +  510-0   +^y=o. 

Note. — The  zero  term  in  (2)  arises  from  the  leverage  of 
the  weight  of  the  bar,  which  is  30  •  o,  zero  being  the  lever-arm. 
But  30*0=0,  for  manifestly,  a 
weight  hanging  at  the  middle  point 
can  have  no  tendency  to  turn  the  bar 
around  this  point,  or,  what  amounts 
to  the  same  thing,  its  turning-tendency 
about  this  point  equals  zero. 

5.  A  stone  slab,  S,  weighing  2,400 
lb.,  rests  with  its  edge  on  a  point, 
B,  6  in.  from  the  fulcrum,  F,  of  a 
crowbar,  FA,  6  ft.  long.  How  many 
pounds  of  force  must  a  man  at  A 
exert  to  raise  the  slab  ? 

Note. — When  the  stone  is  being 
supported,  the  bar,  FA,  is  in  balance  under  three  forces, 
viz.,  the  upward  pressure  of  the   supporting  block   against 


IT 


1 


-IfOf 


Fig.  42 


Equation  Applied  to  Simple  Problems  on  Beams        69 

the  bar,  the  downward  pressure  of  the  stone  at  B  and  the 
upward  pressure  of  the  man's  hand  at  A. 
Show  that  the  equations  are: 

(i)       +/— 2,400+:x;=o,         or /+  :x;=2,4oo; 
(2)   — 3/+6,ooo+3:v=o,     or— 3/+3:v=— 6,000  . 

The  solution  of  this  problem  enables  us  to  find  both  x  and 
/.  In  many  such  problems  the  force  exerted  by  the  support 
enters,  though  it  rarely  needs  computing  because  the  support 
can  easily  be  made  sufficiently  strong.  We  shall  now  solve 
the  problem  without  computing  /.  It  is  clear  that  if  the  bar 
or  other  body  on  which  forces  are  acting  is  balanced,  the  bar 
has  no  tendency  to  turn  around  any  point.  Instead,  then,  of 
choosing  always  the  middle  point,  we  might  take  any  other 
convenient  point. 

If  the  algebraic  sum  of  all  the  turning-tendencies  with 
reference  to  this  point  is  o,  there  is  no  turning  of  the  bar,  or 
the  bar  remains  in  balance. 

It  must  be  noted,  however,  that  once  a  certain  point  is 
selected  from  which  to  measure  lever  arms,  this  point  must 
be  retained  throughout  the  solution. 

To  illustrate:  In  problem  5,  above,  let  all  lever  arms  be 
measured  from  the  point  F;  i.  e.,  choose  F  as  the  turning- 
point.     Equation  (2)  then  becomes: 

(+/)(o)  +  (-2,4oo)(  +  i)  +  (+^)(+6)=o; 
whence  6x— 1,200=0, 

or,  6a; =1,200,     and  finally  :x;= 2 00. 

The  first  equation  thus  becomes  unnecessary. 

It  is  important  to  note  that  by  a  proper  choice  of  the 
turning-point — say  a  point  where  some  force  acts  which  we  do 
not  care  to  know — problems  may  often  be  simplified. 

6.  Three  boys  desire  to  carry  a  12  ft.  log,  weighing  240  lb. 
Two  of  the  boys  lift  at  the  ends  of  a  hand-spike  placed  cross- 


70 


First-Year  Mathematics 


wise  underneath  the  log,  and  the  third  boy  carries  the  rear  end 
of  the  log.  Where  must  the  hand-spike  be  placed  that  all 
may  lift  equally. 

A  glance  at  the  diagram  shows  that  we  have  the  case  of  a 
body  (a  log)  balanced  under  the  effect  of  three  forces.     De- 


0  A 


3 


> * 


i 


y 

Fig.  43 


note  the  force  with  which  each  boy  lifts  by  /,  and  the  distance 
from  the  end,  O,  of  the  log  to  the  point.  A,  where  the  spike  is 
placed,  by  x. 

The  three  forces  are  +2/  at  A,  —240  at  B,  and  +/  at  C. 

The  problem  will  now  be  solved  in  two  ways. 

First,  the  point  O,  will  be  taken  as  the  turning-point. 
By  the  conditions  of  the  problem,  the  lever-arms  will  then  be 
+x,  +6,  and  4-12. 

The  equations  from  Laws  I  and  II  of  forces  are: 

(1)  2/-240-H/  =  0, 

(2)  (-l-2/)(+«;)-h(-24o)(+6)  +  (-h/)(+i2)=o. 

Complete  the  solution. 

Second,  the  middle  point  B,  will  be  taken  as  the  turning- 
point.  Let  y  denote  the  distance  from  the  middle  point,  B, 
of  the  log  to  the  point.  A,  where  the  spike  is  placed  (draw  the 
figure).     The   lever-arms   are   then    —y,  o,  and    -|-6.     The 


Equation  Applied  to  Simple  Problems  on  Beams        71 

first  equation  remains  the  same   as   above,   but  the  second 
becomes: 

(2)    (  +  2/)(-^)  +  (-240)(0)  +  (+/)(  +  6)=0, 

which  reduces  at  once  to  —2Jy+o-\-6j=o, 

or  —  i6o3'+48o=o, 

7.  Solve  problem  6  using  the  end,  C,  as  the  turning- 
point. 

8.  Solve  problem  6  using  A  as  the  turning  point. 

Note. — Observe  that  the  distance  AB  is  d—x.  This  is 
the  lever-arm  of  the  weight  of  the  log  with  reference  to  the 
turning-point  A. 

g.  Suppose  the  length  of  the  log  to  be  /  ft.  and  its  weight 
240  lb.;  find  where  the  spike  should  be  placed  that  each 
boy  may  carry  ^  of  the  weight  of  the  log. 

10.  Suppose  the  length  of  the  log  to  be  /  ft.  and  its  weight 
w  lb.     Find  where  the  spike  should  be  placed. 

11.  If  the  log  be  /  ft.  long  and  the  weight  w  lb.,  where 
must  the  spike  be  placed  if  4  boys,  two  at  each  end  of  the 
spike,  are  to  lift  at  A,  and  a  fifth  at  C,  all  to  lift  equally  ? 

12.  A  steel  beam  24  ft.  long,  and  weighing  120  lb.  per  yd. 
is  being  moved  by  placing  an  axle  borne  by  a  pair  of  wheels 
under  it,  as  shown  at  S^ 

A,  the  end  at  B  being  ^(  .^  '  J^ 

carried.     How   far  ^' 

must    the    a  x  1  e    be  ^ 

placed  from  the  end,  ^^^-  44 

O,  and  what  will  be  the  weight,  W,  on  the  axle,  that  the  weight 
at  B  may  be  200  pounds  ? 
Answer:  5(;=8ff  ft. 

13.  A  steel  rail,  30  ft.  long,  was  supported  at  a  point  14 
ft.  from  one  end.  A  hfting  force  of  45  lb.  at  the  other  end  of  the 
rail  held  the  rail  in  balance.     What  was  the  weight  of  the  rail 


72  First- Year  Mathematics 

per  yd.  of  length?     Find  the  pressure  on  the  supporting- 
point. 

(Draw  the  figure.) 
Answer:   72  lb.;  675  lb. 

14.  If  the  supporting-point  of  the  rail  (problem  12)  had 
been  14^  ft.  from  one  end,  what  force  at  the  other  end  would 
have  balanced  the  rail?  Find  for  this  case  the  pressure  on 
the  supporting-point. 

Answer:   23^x1^.;  696!^  lb. 

15.  How  may  a  railroad  rail  weighing  more  than  a  ton  be 
weighed  with  a  pair  of  balances  running  only  to  60  pounds  ? 

16.  If  the  stone,  S,  presses  down  at  the  point,  P,  with  a 

force  of  4,250  lb.,  and  the 
^     crow-bar,  PA,  6  ft.  long,  is 
supported  at  a  point,  F,  4 
Yj^q  .r  in.  from  the  end,  P,  what 

will  be  the   pressure  at  F 
and  the  downward  force  at  A  when  the  bar  is  balanced  ? 

17.  With  other  conditions  as  in  16,  what  would  be  the  pres- 
sure at  F  and  the  force  at  A,  if  the  distance  from  P  to  F  were 
3  inches  ? 

18.  A  wheelbarrow  is  loaded  with  45  bricks,  averaging 
6  lb.  apiece.  What  Ufting  force 
will  be  needed  at  A  to  raise  the 
load  if  the  bar  OA  is  4^  ft.  and  the 
distance  from  the  center  of  the 
wheel,  O,   to  the  point,  B,  where 

the  vertical  line  through  the  center  of  the  load  crosses  OA, 
is  2  feet  ? 

19.  With  the  same  load  and  length  of  bar,  OA,  as  in  problem 
18,  how  far  is  it  from  O  to  the  crossing -point,  B,  of  the  vertical 
center  line  of  the  load,  if  100  lb.  at  A  just  raises  it  ? 

20.  A  suction  pump  is  a  device  for  raising  water  from  wells. 
The  handle,  OB,  works  against  a  pin  at  A,  so  that  when  the 


Equation  Applied  to  Simple  Prohtems  on  Beams        73 


hand  pushes  downward  at  B,  the  point  O  rises,  and  by  the  aid 
of  a  piston  on  the  lower  end,  C,  raises  a  mass  of  water.  If 
OA  =2  in.  and  OB  =3 
ft.  what  load  at  O  will 
be  raised  by  a  force  of 
20  lb.  pushing  down- 
ward at  B,  and  what 
will  be  the  pressure  on 
the  pin  at  A  ? 

21.  With  conditions 
as  in  20,  what  force  at  O  and  pressure  at  A  will  be  exerted  by 

a  downward  force  of  68  lb.  at  B  ? 

22.  A  dry  goods  box  weighing  360  lb.  is  being  moved  along 
the  floor  by  the  aid  of  a  roller.     If  the  box  is  6  ft.  long  what 


Fig.  47 


Fig.  48 

force  at  C  is  needed  just  to  raise  the  rear  end,  when  the  roller 
is  I  ft.  from  the  front  end  ?  2  ft.  from  the  front  end  ?  5  ft.  ? 
5  in.  ?  6  ft.  ?  6  in.  ?  What  will  be  the  weight  on  the  roller  in 
each  case  ? 

23.  A  foot-bridge  15'  long  between  supports    (15'  span) 

■4. ,si JK 


f 


— s- 


■> 
4' 


-¥ro 


.  J  if  (TV 

FiG.  49 


rests  on  timbers  at  L  and  R.     The  bridge  weighs  i  ,000  pounds. 
Two  men,  whose  combined  weight  is  450  lb.,  stand  just  over 


74  First- Year  Mathematics 

A,  5'  from  the  end  L.  Find  the  pressure  on  the  supports  at 
L  and  R. 

The  bridge  may  be  regarded  as  a  bar  held  in  balance  by 
4  forces:  /  and  r,  called  the  reactions  of  the  supports,  push 
upward  at  the  ends;  the  combined  weight  of  the  two  men 
pushes  downward  at  the  point  5'  from  the  left  end;  and  the 
weight  of  the  bridge  itself  pulls  downward  at  the  middle  of 
the  span. 

In  the  next  two  sections  we  shall  find  r  and  /  first  by  the 
method  of  solving  equations  containing  a  single  unknown  and 
then,  by  the  method  of  two  unknowns. 

§  36.     Solution  by  One  Unknown  Number 

First,  choose  the  turning-point  at  L  and  write  down  the 
equation  of  leverages,  thus: 

(i)(+0(o)  +  (-45o)(  +  5)  +  (-i.ooo)(  +  7i)  +  (+r)(+i5)=o. 
.     Simplifying  this  equation : 

-2,250-7, 500 -|-i5r=o. 
Adding  9,750  to  both  sides  we  have 

(2)  i5r=9,75o,  whence  r  =  65o. 

Therefore  the  upward  pressure  of  the  right  support  against 
the  bridge  is  650  pounds. 

Now  choose  the  turning-point  at  r  and  write  the  equation 
of  leverages,  thus* 

(3)(+0(-iS)  +  (-45o)(-io)  +  (-i,ooo)(-7i)  +  (  +  r)(o) 
=0. 

Simplifying  — 15/4-4,500  +  7,500=0. 

Adding  15/  to  both  sides  and  then  reversing  the  sides  of 
the  equation  we  have: 

(4)  1 5/  =  1 2,000  ;    whence /=8oo. 

Therefore  the  upward  pressure  of  the  left  support  against  the 
bridge  is  800  pounds. 


Equation  Applied  to  Simple  Problems  on  Beams        75 

§  37.     Solution  by  Two  Unknown  Numbers 

To  solve  the  problem  of  §  36  by  the  method  of  two  unknown 
numbers,  both  of  the  laws  of  force  will  be  needed,  the  turning- 
point  being  now  taken  at  the  middle  point,  C,  of  the  span. 

The  equations  furnished  by  the  two  laws  are : 

(i)   +1,-450  —  1,000+22=0  (state  the  law) , 
(2)  (+Z)(-7i)  +  (-45o)(-2i)  +  (-i,ooo)(o) 
+  (+22)(  +  7^)=o  (state  the  law). 

The  latter  equation  becomes 

-7§Z,  +  i,i25  +  7ii?=o. 

Multiplying  this  equation  through  by  2  and  subtracting 
2,250  from  both  sides,  we  find 

i5i2-i5L=-2,25o. 

Dividing  through  by  15,  we  get  72— L=— 150.  Adding  1,450 
to  both  sides  of  the  equation  (i)  gives 

w  1,  ji?+L= +1,450, 

We  now  have  "^  t^     ^ 

|72-L=-i5o. 

Adding  these  two  equations,  member  to  member  (the  sides 
of  an  equation  are  often  called  its  members),  we  have: 

2R  —  1 ,300 ,    or  i?  =  650 . 

Subtracting  the  same  two  equations,  member  from  member, 
we  find: 

2Z,  =  i,6oo,     orZ,=8oo. 

Hence,  the  left  support  presses  upward  against  the  bridge  with 
a  force  of  800  lb.,  and  the  right  with  650  lb.,  as  we  found 
before. 

It  is  well  to  notice  that  by  a  proper  selection  of  the  turning 
point  of  leverages,  the  problem  may  be  solved  by  either  one 
of  the  above  methods. 


76 


First-Year  Mathematics 


1.  If,  in  the  last  example  just  solved,  the  450  lb.  weight 
had  been  6  ft.  from  the  left  support,  what  would  have  been 
the  pressures  on  the  supports  ? 

2.  A  bridge  20  ft.  long  weighs  2,400  lb.,  and  supports 
two  loads;  one  of  600  lb.,  4  ft.  from  the  left  end,  and  the  other 
800  lb.,  15  ft.  from  the  left.  What  are  the  loads  borne  by 
the  supports  ? 

3.  If,  with  the  bridge  of  problem  23,  §  35,  four  loads  of 
450  lb.  each  are  placed,  one  2'  from  the  left  support,  the  second 
6',  the  third  9',  and  the  fourth  16'  from  the  left  end,  what  are 
the  upward  forces  exerted  against  the  ends  of  the  bridge  by 
the  supports? 


A  wagon  box,  EFGO, 


A 

■e- 


3 

■e- 


10'  long  and  loaded  with  40 
bu.  of  wheat  weighing  60  lb. 
per  bu.,  extends  i^'  in  front  of 
the  front  axle,  A,  and  2'  behind 
the  rear  axle,  B.  What  is  the 
load  on  each  axle  ? 


<...i;..> 


-2.'. 


Fig.  50 

5.  If  the  box  (problem  4)  is  filled  with  coal,  it  will  hold 
two  tons.     What  load  on  each  axle  would  this  weight  produce  ? 

6.  A  box  12'  long  of  a  three-horse  coal  wagon  is  loaded 
with  6  tons  of  coal. 
If  the  box  extends 
2'  in  front  of  the 
front  axle  and  4' 
back  of  the  rear  Fig-  Si 

axle,  what  are  the  weights  on  the  front  and  rear  axles  ? 

.  7.  A  lumber  wagon  is  coupled  out  to  a  distance  of  9' 

_ii...i.i_^  between  the  axles 


*K 


-^ 


^'..-Jv ^•- ->  *-3-'- 


— -^- >  and  loaded  with  a 

^x  ^        ^  pile   of    lumber 

f  3i'X4'Xi8'.The 

^^°-  52  load  extends  3'  in 

front  of  the  front  axle  and  the  material  averages  48  lb.  per 


Equation  Applied  to  Simple  Problems  on  Beams        77 


cu.  ft.     What   are   the   pressures   on   the   axles  due  to  this 
load  ?    (Draw  the  figure.) 

8.  The  water  tank  of  a  park  sprinkler  is  a  circular  cylin- 
der 51  in.  in  diameter  and  5 
ft.  high.  A  driver,  weighing 
180  lb.,  sits  at  A,  directly 
above  the  front  axle.  The 
vertical  center  line  through  C 
passes  between  the  wheels  3 
ft.  in  front  of  the  rear  axle, 
and  4  ft.  behind  the  front 
axle.     If  water  weighs  62^  lb.  Fig.  53 

per  cu.  ft.,  what  is  the  weight  on  each  axle,  when  the  tank 
is  level  full  and  the  driver  is  on  the  seat  ? 

9.  A  wagon  standing  on  a  culvert,  AB,  has  on  its  front 

axle  a  load  of  2,500 
lb.,  and  on  its  rear 
axle,  3,000  lb.  The 
front  wheels  are  4' 
and  the  rear  wheels 
10'  from  the  left  end 
of    the    culvert.     If 

what   are    the    pressures    at    the 


<- >»- 


FT 


-20- 
-> 


^ 


Fig.  54 
20'   long. 


7 


the   culvert    is 
supports  ? 

10.  What  would  these  pressures  be  if  the  front  wheels  stood 
9'  from  the  left  support  (wagon  coupled  to  6'  between  the 
axles)  ? 


Fig.  55 


CHAPTER  XII 

THE  SIMPLE  EQUATION 

§  38.     The  Axioms 

A  light  bar,  AB,  is  carefully  balanced  at  C.    Equal  weights, 

w,  w',  are  then  hung  at 
equal  distances  on  op- 
posite sides  of  C,  as  shown 
■g  in  Fig.  55.  The  beam 
remains  balanced  under 
these  weights. 

I.  Two  4-0Z.  weights 
are  now  hung  to  w.  How  many  ounces  must  be  hung  to  w' 
for  balance  ? 

2.  If  another  4-oz.  weight  is  hung  to  w,  how  many  ounces 
must  be  hung  to  w'  for  balance  ? 

3.  If  5C  oz.  are  hung  to  w,  how  many  ounces  must  be  hung 
to  w'  for  balance  ? 

Show  that  these  problems  illustrate  the  truth  of  the 
following : 

Addition  Axiom. — //  the  same  number,  or  equal  numbers, 
be  added  to  equal  numbers,  the  sums  are  equal. 

4.  Hang  six  4-oz.  weights  on  w.  How  many  2-oz.  weights 
must  be  hung  on  w'  for  balance  ? 

5.  If  a  4-0Z.  weight  is  now  removed  from  the  left  side  how 
many  2-oz.  weights  must  be  taken  off  the  right  side  to  restore 
the  balance  ? 

6.  If  two  more  4-oz.  weights  are  taken  from  the  left,  how 
many  2-oz.  weights  must  be  taken  from  the  right  for  balance  ? 

7.  If  the  bar  is  balanced  at  first  by  a  pair  of  equal  weights 
of  any  number  of  pounds  and  any  number  of  pounds  is  taken 

78 


The  Simple  Equation  79 

from  the  left,  how  many  pounds  must  be  taken  from  the  right 
for  balance  ? 

8.  If  the  bar  is  balanced  under  x  lb.  on  each  side  of  C,  and 
y\h.  is  taken  from  the  left,  how  many  pounds  must  be  taken 
from  the  right  for  balance  ? 

What  has  been  suggested  in  the  last  problem  may  be 
written  thus: 

(i)  x=x 

(2)  y=y 

(3)  x-y=x-y. 

Show  that  the  last  five  problems  illustrate  the  following: 
Subtraction  Axiom. — //  the  same  number,  or  equal  num- 
bers, be  subtracted  from  equal  numbers,  the  differences  are  equal. 

9.  Suppose  the  bar  of  Fig.  55  is  first  balanced  by  a  2-oz. 
weight  on  each  end.  If  another  2-oz.  weight  is  added  on  the 
left,  how  many  2-oz.  weights  on  the  right  will  balance  the 
bar? 

10.  Suppose  the  bar  is  balanced  under  any  pair  of  equal 
weights.  If  the  weight  on  the  left  is  doubled  what  change 
must  be  made  on  the  right  for  balance  ?  What  change  must 
be  made  on  the  right  if  the  weight  on  the  left  is  trebled? 
quadrupled  ?  What  change  must  be  made  on  the  right  if 
the  weight  on  the  left  is  multiplied  by  6?  by  12?  by  25? 
by  a? 

Show  how  these  questions  illustrate  the  truth  of  the  fol- 
lowing : 

Multiplication  Axiom. — //  equal  numbers  be  multiplied 
by  the  same,  or  equal,  numbers,  the  products  are  equal. 

In  problems  11,  12,  and  13,  suppose  the  bar  of  Fig.  55 
is  balanced  by  12  2 -lb.  weights  on  each  end. 

11.  If  half  as  many  weights  are  placed  on  the  left,  what 
part  of  the  number  of  weights  on  the  right  must  remain  for 
balance  ? 


8o  First-Year  Mathematics 

12.  If  J  of  the  weights  are  on  the  right,  what  part  must 
remain  on  the  left  for  balance  ? 

13.  What  part  on  the  right  must  remain  for  balance,  if 
\  of  those  on  the  left  remain  ?    if  ^  ?    if  tV  '^ 

These  exercises  make  clear  the  following  axiom. 
Division  Axiom. — //  equal  numbers  he  divided  by  the  same 
number,  or  by  equal  numbers,  the  quotients  are  equal. 
Exercises  on  the  axioms. 

14.  The  bar  is  balanced  under  t^x  lb.  on  one  side  and  9  lb. 
on  the  other.  Six  pounds  additional  are  now  hung  to  each 
side  and  the  bar  still  balances.  This  is  expressed  in  algebraic 
language  thus: 

(i)  3^=9 

(2)  6   =6     (Any  number  may  be  written  equal  to  itself.) 

2,x-{-6  =  is. 

15.  If  x=%  and  a  =  6  what  third  relation  may  be 
written  from  the  addition  axiom  ?  What  from  the  subtrac- 
tion axiom?  The  multiplication  axiom?  The  division 
axiom? 

16.  If  4;x;— 5  =  7  and  we  write  5  =  5,  what  third  relation 
may  be  written  from  the  addition  axiom?  From  the  multi- 
plication axiom  ? 

17.  If  :x:— 11=10,  show  how  to  find  the  value  of  x  by  the 
addition  axiom. 

18.  If  a;+5=9,  show  how  to  find  the  value  of  x  by  the 
subtraction  axiom. 

19.  If  431; =12,  show  how  to  find  the  value  of  x  by  the 
division  axiom. 

20.  If  ^x=6  ,  show  by  the  multiplication  axiom  how  to 
find  what  x  equak. 

21.  Give  the  reason  for  the  truth  of  the  third  statement  in 
each  of  the  following  cases: 


The  Simple  Equation  8i 

If,  0=  9  and   6=3,  then   a+h  =  i2  ; 

w  =  i3  and  n=2  ,  then     ww=26  ; 

a=  X  and  h  =  y ,  then  fl+6=:x;+3' ; 

c  =  i6  and  (i=4,  then  c—d— 12  ; 

w  =  28and«  =  7,     then —=4  ; 

c  .   d  .        c+d 

—  =  "5  and —  =  7,,    then    =8. 

12     ^  12     ""  12 

22.  Give  the  reasons  for  the  following: 

If  a=7,      50=35;  If      2^=6,     ^=3; 

^=3  J       /=48;  ^-5=3 »    ^=8; 

f»w=6w,    w=  6;  —  =  7,    r=7o. 

10 

§  39.     Solution  of  Equations  by  Axioms 

1.  If  ^x+g=^x+i'j,  find  the  value  of  x. 
Solution. — 

5-'v+9=3^+i7 

3^       =3X  (Any  number  may  be  written 

/ .     2^+9  =  17  ^Why  ?         ^qu^l  to  itself.) 

9=  9  Why? 

.-.     2x=  8  ^Why  ? 

:»=  4  Why? 

and  the  value  of  x  is  4. 

Check:  5  •  4  +  9=3  *  4  +  17=29  . 

2.  5:x;— 17+3X— 5=6:x;  — 7— 85(;+ii5.     Find  jc. 
Solution. — Collecting  Hke  terms  on  each  side, 

8rx;— 22  =  — 23f+io8 

22  =  22  Why? 

8x=—2x-\-i30    Why? 
2r!c=2:!i(;  Why? 


ioj£;=i30  Why? 

and      x=i2  Why? 

Check  this  result. 


82  First-Year  Mathematics 

Solve  the  following  equations  for  the  values  of  x  and  check 
all  results: 

3.  3:^+15=^+25. 

4.  2x-s=sx-T. 

5.  7^—39-10^+15  =  100-33^+26. 

6.  118— 6531;- i23  =  i5:x;+35  — 120JC. 

■ix 

7-  =y  +  5=9i-io^- 
4 

8.  8  +  2:;e+-  =  if+-. 

4  3 

9.  3(^-2)+i5  =  5^-3- 

10.  3JC— 2(.r+5)=6^— 20. 

11.  5(:x;-i)  =3(^+1). 

12.  3X-\-i4-s{x-3)=4(x-\-3)- 

13.  2{x-i)+3{x-2)=4{x-5). 

14.  2(:x;+i)+3(:x;+2)+4(5£;+3)=iio. 

15.  s(x-2)+6(x-i)-4(x-s)=()0. 

16.  2(r)(;-3)+3(2x+i)-2(35(;-i)=47. 

17.  4(2^+9)+3(^-9)-S(2^-7)  =  i3- 

18.  x(x—i)—x(x—2)=4(x—3). 


CHAPTER  XIII 

THE  GRAPH 

§  40.     Locating  Points  by  Means  of  Numbers 

Choose  some  intersection  on  a  piece  of  squared  paper  to 
represent  the  position  of  the  central  point  of  a  city  and  let  one 
small  square  represent  one  city-block.  From  this  fixed  ref- 
erence point,  or  origin,  let  the  position  of  an  object  m  blocks 
east  (to  the  right  on  the  paper)  be  denoted  by  +wE,  and 
m  blocks  west  (to  the  left)  by  —  wE.  Then  if  an  object  is 
situated  at  +4E,  it  is  somewhere  on  the  street  4  blocks  east  of 
the  central  point  or  4  blocks  east  of  the  north  and  south  street 
through  the  central  point.  We  will  call  the  north  and  south 
street  through  the  origin  the  N-S  axis,  and  the  east-west 
street  through  the  origin  the  E-W  axis.  Similarly,  an  object 
at  —  2E  is  somewhere  on  the  street  2  blocks  west  of  the  origin, 
or  of  the  N-S  axis.  The  exact  north  and  south  positions  of 
the  object  are,  however,  not  given  in  either  case. 

1.  Let  the  position  of  an  object  k  blocks  north  (upward 
on  the  paper)  be  denoted  by  -1-^N;  and  k  blocks  south  (down- 
ward on  the  paper),  by  —  ^N.  Then  if  an  object  is  situated 
at  —  7N,  it  is  somewhere  on  the  street  7  blocks  south  of  the 
origin,  or  the  E-W  axis,  and  at  +5N,  it  is  somewhere  on  the 
street  5  blocks  north  of  the  origin,  or  the  E-W  axis.  In  this 
case  the  exact  east  and  west  positions  of  the  objects  are  un- 
known. Point  out  the  position  of  an  object  at  -}-3E  and  at 
the  same  time  at  —  7N  (See  Fig.  56,  p.  84).  How  many 
points  are  so  situated  ? 

2.  Point  out  the  positions  of  the  following  places: 


+IIE, 

-7N 

+  2E, 

+9N; 

+3iE, 

+  SN 

+  2E, 

-9N; 

-SE    , 

-6N 

-2E, 

+9N; 

-4E    , 

-3N 

-2E, 

-9N. 

83 


84 


First-Year  Mathematics 


3.  If  it  is  agreed  that  the  E-distance  shall  always  be  written 
first,  and  the  N-distance  second,  the  position  of  a  point  may  be 


/V 


o 


\0£cJ^l 


Fig.  56 


represented  by  two  numbers  with  their  signs;    i.  e.,   +11E, 
— 7N  may  be  written  simply  +11,  —7. 
Locate  the  following  points: 


The  Graph 


85 


+3, 
+9, 


+  7>    -S;  -  8,    -  3 ;  0,-4; 

-  6,    -4-;  -II,    +11  ;  o,    +3  ; 

-13,    +9  ;  +  6,    +  3  ;  -3,       o; 

+  4,4-6;  +  8,    -  7  ;  o,       o. 

4.  Locate  the  following  points: 
+  1  ;  +12,    +4;  -3,    -I  ;  +6,    +2; 
+3;          +18,    +6;             o,       o;         -6,    -2. 

5.  How  do  the  points  in  problem  4  He  with  respect  to  each 
other  ?  What  is  the  relation  between  the  E-distance  and  the 
N-distance  for  all  the  points.  Draw  the  line  passing  through 
the  points.  Take  some  other  points  on  the  line  and  measure 
their  co-ordinates  (i.  e.,  their  E-distance  and  their  N-distance). 
Do  their  values  have  the  same  ratio  to  each  other  as  the  co- 
ordinates of  the  points  in  problem  4  ? 

§  41.     Picturing  or  Plotting  Laws  Connecting  Numbers 

I.  Find  ten  points  for  which  N  =  5E  and  locate,  or  plot, 
their  positions  on  the  squared  paper.  How  do  they  lie  with 
respect  to  each  other?  Make  a  table  of  the  co-ordinates  of 
the  points  as  follows: 


E 

N 

—  2 

— I 

0 

+  1 

—  10 

-  5 
0 

+  5 

2.  In  the  relation  N  =  2E-h5,  let  E  take  all  the  integral 
values  (whole  numbers)  from  —7  to  +7.  Make  a  table  of  the 
points  and  plot  them.  Do  they  all  lie  in  a  single  straight 
line?  What  is  the  least  number  of  points  from  which  the 
position  of  such  a  straight  line  may  be  fixed  ? 

Note. — A  hne   determined  as  in  problem   2    from 
equation  is  said  to  represent  the  equation. 


an 


86  First- Year  Mathematics 

3.  Plot,  from  three  points  (two  to  fix  the  position  of  the 
line,  and  the  third  as  a  check)  the  straight  lines  which  repre- 
sent the  following  equations: 

(i)N  =  s-2E;  N=3i:4E 

(2)  N=iE;  ^^^  2      ' 

(,)N  =  f+5-  (5)2N+5E  =  4; 

^3^  ^     ^+5  '  (6)  2N-3E  =  s. 

4.  In  each  part  of  problem  3  measure  the  E-distance  from 
the  origin  to  the  point  where  the  graph  of  the  equation  cuts 
the  E-W  axis.    What  is  the  value  of  N  for  this  point  ? 

It  is  evident  that  for  N=o  in,  say,  problem  3, 

(i)  5-2E=o, 

and  E  =  2^. 

Thus  from  the  graph  of  the  equation  N  =  5  — 2E,  we  may 
obtain  the  solution  of  the  equation  5  — 2E=o. 
For  example,  to  solve  the  equation 

^+ii=4E  +  i9, 

transpose,  or  change  all  terms  to  the  right  side  of  the  equa- 
tion by  means  of  the  axioms  and  simpUfy,  whence 

o=5E-|-i6. 

Put  the  expression  on  the  right  equal  to  N,  thus: 

N  =  5E  +  i6. 

Plot  the  graph  of  this  equation  and  measure  the  distance 
from  the  origin  to  the  point  where  the  graph  cuts  the  E-W 
axis.    This  gives  the  value  of  E  for  which  N=o,  or  for  which 

5E-l-i6=o, 

i.  e.,  we  get  the  solution  of  the  equation 

^  +  ii=4E-hi9. 

This  method  of  solution  is  called  the  graphical  method. 


The  Graph  87 

5.  Solve  the  following  by  the  graphical  method: 

(1)  ^  +  2E-9=-V-  +  sE; 

(2)  5E  +  ii+9E-2  =  6E-s; 

(3)  4(3-2E)=7(iiE-5); 

(4)  5(2E-4)+8(E-4)=i7; 

(5)  3(E-3)-5(2E-s)=9(E  +  2)-4(2E+3). 

6.  Suppose  the  E-W  axis  is  called  the  :x;-axis  and  the  plus 
and  minus  E-distances  are  called  plus  and  minus  :v-distances, 
and  the  N-S  axis  is  called  the  )'-axis  and  the  plus  and  minus 
N-distances  are  called  plus  and  minus  ^'-distances.  Assum- 
ing the  first  number  in  each  pair  to  be  the  ic-distance,  plot  the 
following  points: 

-5.  +  4;  -7,  +15;  -9,-8;  o,      o; 

-3,  +11  ;  +5,  +12  ;  +8,. -i;  0,-4. 

7.  Tabulate  from  the  following  equations  the  values  of  y 
for  all  integral  values  of  x  from  —5  to  +5;  plot  the  graph, 
and  measure  the  value  x  for  which  y  =0: 

(i)  y=^-s  ;  (3)  y=5(7-4^)-i8; 

(2)  y=2>{2X-s)\  (4)  5)'=6(2-3^)-5(i-:x;). 

8.  Solve  graphically  the  following  equations,  and  check 
each  result  by  solving  algebraically  by  means  of  the  axioms: 
(i)  2^+3  =  i6-(2:x;-3);  (3)  2^(7-3 (2:^-3)  =i-4(j£;- 2); 

(2)  8(10-:.)  =  5(^+3)  ;  (4)  y-^=lf  . 


.      CHAPTER  XIV 

EQUATIONS  CONTAINING  FRACTIONS 
§  42.     Freeing  an  Equation  of  Denominators 

1 .  To  find  the  value  of  x  in  an  equation  of  the  form 

it  is  necessary  to  clear  the  equation  of  fractions  by  multiplying 
both  sides  of  the  equation  by  the  denominator  7  (multiplica- 
tion axiom) ;  then 

;  ,;:  and  :!C  =  28. 

Sx 
Find  X  in  —  =  16. 
9 

2.  If  the  equation  involves  several  fractions  it  may  be  sim- 
plified by  multiplying  both  sides  of  the  equation  by  a  number 
which  will  contain,  as  factors,  all  of  the  denominators;  e.  g., 
to  solve 

X     2X     sx_ 

both  sides  of  the  equation  may  be  multiplied  by  70,  giving 

55x-2Sx-\-iox=y4, 
whence  37^=74  > 

and  x=  2 . 

_,.   J      .    X     ix     gx 
348 

3.  In  simphfying  an  equation  involving  several  fractions 
it  is  best  to  multiply  both  sides  of  the  equation  by  the  least 
number  which  contains,  as  factors,  all  of  the  denominators; 
i.  e.,  by  the  least  common  multiple  of  the  denominators,  called 


Equations  Containing  Fractions  89 

the  least  common  denominator.  The  least  common  denomi- 
nator is  found,  as  in  arithmetic,  by  taking  the  product  of  the 
different  prime  factors  of  all  the  denominators,  each  factor 
being  used  the  greatest  number  of  times  it  occurs,  as  a  factor, 
of  any  one  denominator;  e.  g.: 
X      sx    iix 

24=2X2X2X3, 
42=2X3X7, 
56=2X2X2X7. 
Two  occurs,  at  most,  three  times;   three,  at  most,  twice; 
and  seven,  at  most  once,  in  the  factors  of  any  one  denominator, 
therefore  23X3^X7  is  the  least  common  multiple  of  the  num- 
bers 24,  56,  and  36,  or  the  least  common  denominator  of  the 
given  denominators. 

Solution. — Multiply  both  sides  of  the  equation  (i)  by 
23X3'X7,  and 

2i:x;—6o:v-|-995(;= 14X17, 
complete  the  solution. 

It  is  advisable  in  clearing  equations  of  fractions  to  avoid 
multiplying  out  results  until  absolutely  necessary. 

Query. — Could  an  equation  be  freed  of  its  denominators 
by  multiplying  both  sides  by  any  other  number  or  numbers, 
than  the  least  common  multiple  of  the  denominators  ? 

§  43.     Problems  in  Equations  Involving  Fractions 
1.  Simplify  the  following  equations  and  find  the  value  of  x: 
(  \  ^^     5     ^ . 


(2) 


*-8      , 


X-^       X+T, 

(3)  — +^+Tft:=o; 


90 


First-  Year  Mathematics 


x-zx-^S     X+2_     , 


vS;  — : — r— — ri——- — iiTF> 


(6) 


2JC-5      a;-3  ^4Jg-3 

5        25f-i5       lo 
4(^  +  3)     8:v-t  37     7:x;— 29 


9  18         5X- 


'     2ic+3     4X— I        ' 

5^-3    5^+2 

, .      I        6    6+x 
(9) \—  = . 

l—X      X      J—X 

2.  Solve  the  following  literal  equations: 
,  ,   x—h    x—a 

Solution. — Multiply  both  sides  by  ab,  the  least  common 

multiple  of  the  denominators,  and  obtain 

hx—h'=ax—a'^. 

Transpose  the  terms  containing  x  to  the  left  side  and  all 

other  terms  to  the  right  side  of  the  equation: 

hx—ax=h^—a^, 

or,  x(b—a)=b'—a'         (since  (b—a)x=bx—ax) , 

dividing  by  (b—a),  x=b+a  (since  (6+a)(6— a)=6"— a^)  . 

, ^  X ,   X  , .   x+m     , 

(2)  -H —  =  3  ;  /.^  a  - 

^      a    2a 

.  .  mx    nx       ,  ,    , 

(4)  — H =w»+»*  ; 

n      m 

^^^  wTx    n    x' 

(6)  cx+a-\--=--  ; 
^  c    a 


^^'  x-n-^' 

(8)      1    =     ^     • 
m+x    m—x 

(9)  Solve  (i)  for  a  ; 

iorb. 

(10)  Solve  (2)  for  a . 

(11)  Solve  (4)  for  m 

;  for  n 

(12)  Solve  (8)  for  m. 

Equations  Containing  Fractions  91 

§  44.     Problems  Leading  to  Equations   Containing  Fractional 
and  Literal  Numbers 

1.  Bell  metal  is  by  weight  five  parts  tin  and  16  parts  cop- 
per. How  many  pounds  of  tin  and  of  copper  are  there  in  a 
bell  which  weighs  4,200  lb.  ? 

Solution. — First  method:  Let  x  be  the  number  of  lb.  of  tin, 

then  —  is  the  number  of  pounds  of  copper. 

Ihen,  x-\ =4,200, 

5 
whence,  x=i  ,000 , 

,  rbx 

and  =^,200. 

5      ^ 
Therefore  there  are  1,000  lb.  of  tin  and  3,200  lb.  of  copper. 

Second  method  (without  fractions) :  Let  ^x  be  the  number 
of  lb.  of  tin,  then  itx  is  the  number  of  lb.  of  copper; 

so  that  5x+i6:)(;=4,2oo, 

whence  rx;=    200 . 

Then,  5:x;= 1,000, 

and  i65(;=3,2oo. 

Therefore  there  are  1,000  lb.  of  tin  and  3,200  lb.  of  copper. 

2.  Gunpowder  contains  by  weight  6  parts  saltpeter,  i 
part  sulphur,  and  i  part  charcoal.  How  many  pounds  of 
saltpeter,  of  sulphur,  and  of  charcoal  are  there  in  120  lb.  of 
gunpowder  ? 

3.  A  certain  compound  contains  by  weight  5  parts  of  car- 
bon to  every  3  parts  of  iron,  and  7  parts  of  iron  to  every  2  parts 
of  copper.  In  124  lb.  of  the  compound  how  many  pounds 
are  there  of  carbon,  of  iron,  and  of  copper  ? 

4.  In  an  alloy  of  silver  and  copper  weighing  90  oz.  there 
are  6  oz.  of  copper;  find  how  much  silver  must  be  added  so  that 
10  oz.  of  the  new  alloy  shall  contain  f  of  an  ounce  of  copper. 

5.  If  80  lb.  of  sea- water  contain  4  lb.  of  salt,  how  much 
fresh  water  must  be  added  to  make  a  new  solution  of  which 
45  lb.  contain  f  lb.  of  salt  ? 


g2  First-Year  Mathematics 

6.  One  bin  contains  a  mixture  of  14  bu.  of  oats  and  16 
bu.  of  wheat.  Another  contains  20  bu.  of  oats  and  12  bu.  of 
wheat.  How  many  bushels  must  be  taken  from  each  bin  to 
make  a  mixture  of  5  bu.  of  oats  and  5  bu.  of  wheat  ? 

7.  If  I  lb.  of  lead  loses  ^  of  a  lb.,  and  i  lb.  of  iron  loses 
■^  of  a  pound,  when  weighed  in  water,  how  many  pounds  of 
lead  and  of  iron  are  there  in  a  mass  of  lead  and  iron  that 
weighs  159  lb.  in  air  and  143  lb.  in  water? 

8.  If  a  mass  of  gold  weighs  97  oz.  in  air  and  92  oz.  in  water, 
and  a  mass  of  silver  weighs  21  oz.  in  air  and  19  oz.  in  water, 
how  many  ounces  of  gold  and  of  silver  are  there  in  a  mass  of 
gold  and  silver  that  weighs  320  oz.  in  air  and  298  oz.  in  water  ? 

9.  Into  what  two  sums  can  $1,000  be  divided  so  that  the 
income  of  one  at  6  per  cent,  shall  be  equal  to  the  income  of  the 
other  at  4  per  cent.  ? 

10.  What  percentage  of  evaporation  must  take  place 
from  a  6  per  cent,  solution  of  salt  and  water  (salt  water 
of  which  6  per  cent,  by  weight  is  salt)  to  make  the  remaining 
portion  of  the  mixture  an  8  per  cent,  solution?  12  per  cent, 
solution  ? 

n.  A  physician  having  a  6  per  cent,  solution  of  a  certain 
kind  of  medicine  wishes  to  dilute  it  to  a  3^  per  cent,  solution. 
What  percentage  of  water  must  be  added  to  the  present  mixture  ? 

12.  An  express  train  whose  rate  is  40  mi.  per  hour  starts 
I  hr.  and  4  min.  after  a  freight  train,  and  overtakes  it  in  i  hr. 
and  36  minutes.  How  many  miles  does  the  freight  train  run 
per  hour  ? 

Suggestions. — Let  x  be  the  rate  of  the  freight  train  in 
miles  per  hour.  Both  trains  travel  the  same  distance, 
therefore  2fX^=4oXif , 

9>x    ^ 

5i;=24. 
Hence  the  freight  train  runs  24  miles  per  hour. 


Equations  Containing  Fractions  93 

13.  Two  trains  start  at  the  same  time  from  S,  one  going 
east  at  the  rate  of  35  mi.  per  hr.  and  the  other  going  west  at 
a  rate  \  faster.  How  long  after  starting  will  they  be  exactly 
100  mi.  apart  ? 

14.  A  man  walks  beside  a  railway  at  the  rate  of  4  mi.  per 
hr.  If  a  train  208  yd.  long,  traveling  30  mi.  per  hr.,  overtakes 
him,  how  long  will  it  take  the  train  to  pass  the  man  ? 

15.  A  railroad  train  moves  at  a  uniform  rate.  If  the  rate 
were  6  mi.  per  hr.  faster,  the  distance  it  goes  in  8  hr.  would 
be  50  mi.  greater  than  the  distance  it  would  go  in  11  hr.  at 
a  rate  7  mi.  per  hr.  less  than  the  actual  rate.  Find  the  actual 
rate  of  the  train. 

16.  Two  trains  go  from  P  to  Q  on  different  routes,  one 
of  which  is  15  miles  longer  than  the  other.  A  train  on  the 
shorter  route  takes  6  hr.  and  a  train  on  the  longer,  running 
10  miles  less  per  hr.,  takes  8^  hr.  Find  the  length  of  each 
route. 

17.  The  distance  from  A  to  B  is  100  miles.  A  train  leaving 
A  at  a  certain  rate  meets  with  an  accident  20  miles  from  B, 
reducing  its  speed  one-half,  and  causing  it  to  reach  B  an  hour 
late.     What  was  its  rate  per  hour  before  the  accident  ? 

18.  A  man  rows  down  stream  at  the  rate  of  6  mi.  per  hr. 
and  returns  at  the  rate  of  3  mi.  per  hr.  How  far  down  stream 
can  he  go  and  return  within  9  hr.  ? 

19.  A  boatman  moves  5  mi.  in  f  of  an  hour,  rowing  with 
the  tide;  to  return  over  the  same  route  it  takes  him  i^  hr., 
rowing  against  a  tide  one-half  as  strong.  What  is  the  velocity 
of  the  stronger  tide  ? 

20.  A  boatman,  rowing  with  the  tide,  moves  a  mi.  in  h 
hours.  Returning,  it  takes  him  chr.  to  cover  the  same  distance, 
rowing  against  a  tide  m  times  as  strong  as  the  first.  What 
is  the  velocity  of  the  stronger  tide  ? 

21.  The  distance  from  A  to  B  is  32  miles.  A  man 
sets  out  from  A  and  reaches  B  12  minutes *^after  a  second 


94  First- Year  Mathematics 

man,  who  left  A  when  the  first  man  had  proceeded  ii  miles 
upon  his  journey.  If  the  second  man  can  make  the  journey 
in  4  hours,  at  what  distance  from  B  did  he  pass  the  first  man  ? 

22.  A  and  B  run  a  race.  At  the  end  of  5  minutes  when 
A  has  run  900  yd.  and  is  ahead  of  B  by  75  yd.,  he  falls 
and  for  the  rest  of  the  race  makes  20  yd.  less  per  minute  than 
at  first.  He  comes  in  one-half  minute  behind  B.  What  was 
B's  time  ? 

23.  A  bicyclist  traveUng  a  mi.  per  hr.  is  followed,  after  a 
start  of  m  miles  by  a  second  bicychst,  traveUng  b  mi.  per  hr., 
(6>a).  At  these  rates,  in  how  many  hours  after  the  second 
starts  will  he  overtake  the  first  ? 

24.  At  what  time  between  three  and  four  o'clock  are  the 
hands  of  the  clock  together? 

Suggestion. — Let  x  represent  the  number  of  minute 
spaces  over  which  the  minute  hand  passes  from  three  o'clock 

X 

on  until  it  first  overtakes  the  hour  hand;  then  show  that f-iS 

represents  the  same  number  of  spaces;  whence  the  equation  is: 

Hence  the  hands  are  together  at  i6y\  minutes  past  3  o'clock. 

25.  At  what  time  between  two  and  three  o'clock  are  the 
hands  of  the  clock  together  ? 

26.  At  what  time  between  3  and  4  o'clock  are  the  hands 
of  the  clock  at  right  angles  (two  results)  ? 

27.  At  what  time  between  7  and  8  o'clock  are  the  hands 
of  the  clock  pointing  in  opposite  directions  ? 

28.  At  what  time  between  5  and  6  o'clock  are  the  hands  of 
the  clock  one  minute  space  apart  ?     (Two  results.) 

29.  If  the  second  hand  of  a  watch  has  made  ^j  of  a  revolu- 
tion and  the  minute  and  hour  hands  of  the  watch  are  together, 
what  time  is  it  ? 


Equations  Containing  Fractions  95 

30.  The  planet  Venus  passes  around  the  sun  13  times  to 
the  earth's  8.  How  many  months  is  it  from  the  time  when 
Venus  is  between  the  earth  and  the  sun  to  the  next  time 
when  it  is  in  the  same  relative  position  ? 

31.  Seen  from  the  earth,  the  moon  completes  the  circuit 
of  the  heavens  in  27  da.,  7  hr.,  43  min.,  4.68  sec,  and  the 
sun  in  365  da.,  5  hr.,  48  min.,  47 .8  sec,  in  the  same  direction. 
Required  the  time  to  o .  0001  da.  from  one  full  moon  to  the  next, 
the  motion  supposed  to  be  uniform. 

32.  The  difference  between  the  squares  of  two  consecutive 
numbers  is  19.     Find  the  numbers. 

33.  A  box  of  oranges  was  bought  at  the  rate  of  15  cents  a 
dozen.  Five  dozen  were  given  away  and  the  remainder  sold 
at  the  rate  of  2  for  5  cents.  If  this  gave  a  profit  of  30  cents 
on  the  box,  how  many  were  there  in  the  box  ? 

34.  A  company  of  men  was  drawn  up  in  a  hollow  square 

4  deep.  Afterward  it  was  separated  into  two  detachments. 
One  was  a  hollow  square  3  deep  and  with  34  more  men  in 
front  than  formerly,  and  the  other  contained  80  men.  How 
many  men  in  the  company  ? 

35.  A  man  engaged  to  work  20  days  on  these  conditions: 
for  each  day  he  worked  he  was  to  receive  2  dollars,  and  for 
each  day  he  was  idle  he  was  to  forfeit  i  dollar.  At  the  end 
of  20  days  he  received  34  dollars.  How  many  days  was  he 
idle? 

36.  A  man  engaged  to  work  a  days  on  these  conditions: 
for  each  day  he  worked  he  was  to  receive  h  dollars;  and  for 
each  day  he  was  idle  he  was  to  forfeit  c  dollars.  At  the  end 
of  a  days  he  received  d  dollars.  How  many  days  was  he 
idle  ? 

37.  The  circumference  of  the  fore  wheels  of  a  carriage  is 
9  ft.;  that  of  the  hind  wheels,  12  feet.  What  distance  will 
the  carriage  have  passed  over  when  the  fore  wheels  have  made 

5  more  revolutions  than  the  hind  wheels  ? 


96  First-Year  Mathematics 

38.  The  circumference  of  the  fore  wheels  of  a  carriage  is 
a  ft.;  that  of  the  hind  wheels,  h  ft.  What  distance  will  the 
carriage  have  passed  over  when  the  fore  wheels  have  made 
n  more  revolutions  than  the  hind  wheels  ? 

39.  A  man  spends  one  ath  of  his  income  for  food,  one  h\h. 
for  rent,  one  cth  for  clothing,  one  dth.  for  furniture  and  he  saves 
e  dollars.    How  much  is  his  income  ? 


CHAPTER  XV 

THE    FUNDAMENTAL    PROCESSES    APPLIED    TO    INTEGRAL 
ALGEBRAIC   EXPRESSIONS 

§  45.     Addition  of  Integral  Algebraic  Expressions 

An  algebraic  expression  whose  parts  are  numbers  connected 
by  addition  or  subtraction,  or  by  both,  is  called  a  polynomial. 
The  parts  of  the  polynomial,  together  with  the  signs  preced- 
ing them,  are  called  terms. 

(I  ic 

How  many  terms  are  in  --\-Kax-\-~  ? 
b  m 

How  many  terms  are  in  —6  +  ^x-{-'jy-7-2X—4? 

If  a  term  is  composed  of  factors,  any  one  of  the  factors, 

or  the  product  of  several  factors,  is  called  the  coefficient  of 

the  product  of  the  other  remaining  factors;  e.  g.,  in  the  term  ^ax 

the  coefficient  of  x  is  3a , 

"  ax  is  3, 

"  sxisa, 

"  sis  ax. 

What  is    the   coefficient    of    b  in    s^bx,    2abcx,    ^abxy, 

—  6abxz  -,     ^  -      ,     .     ,  ... 

r     Of   abx  m  the  same  expressions  ? 

gc 

What  is  the  coefficient  of  ax  in  the  same  expressions  ? 

Show  by  counting,  of  units  that  2a4-3a  =  5a,  $a  +  2a-\-'ja  = 
14a. 

Unite  3:x;-|-5:x;  into  one  term;    also  jy+gy—sy- 

Give  a  single  term  equal  to  xy+xy,  or  ^x—^x,  or  $xy+ 
8xy—2xy,  or  2(x—y)+4(x—y)—s(x—y) . 

Can  Jx+§>'  be  united  into  one  term  ? 

Under  what  condition  is  it  possible  to  unite  terms  into  a 
single  term?  Make  a  rule  for  adding  numbers  of  the  same 
denomination. 

97 


gS  First- Year  Mathematics 

1.  Express  the  following  in  as  few  terms  as  possible: 
(i)  7a  +  (-5a);  (6)  2o^+(-25x); 

(2)  ioa  +  (-7a);  (7)   -36  + (-4a); 

(3)  i6y+(-9y);  (8)  7m  +  (-i6/')  +  (-3w)  +  (-ia/>); 

(4)  — 8a  +  i7a;  (9)  i6:x:+2i:x;+(  — ii:v); 

(5)  -9^+12^;         (10)  -ly+i-^y). 

2.  To  6  times  a  given  number  add  12  times  that  number, 
then  add  to  this  sum  —  7  times  the  given  number.  What  is 
the  result  ? 

3.  The  lengths  of  two  lines  are  <,i  and  34.  What  is  their 
sum  ?     Find  the  sum  both  algebraically  and  geometrically. 

4.  A  man  mixes  27^  of  saltpeter  with  i2g  of  sulphur  and 
i2gfii  charcoal.     How  much  gunpowder  does  he  get  ? 

5.  The  property  of  a  man  consists  of  2oo:»  acres  in  fields, 
8o3f  acres  in  meadow,  60a;  acres  of  woodland,  and  2o:x;  acres 
of  garden.     How  many  acres  does  he  own  in  all  ? 

6.  Add  the  following  numbers: 


(I 
(2 
(3 
(4 
(5 
(6 
(7 

(s; 

(9: 

(10; 

(" 
(12 

(13 
(14: 
(15 
(16: 


6»,  7»,  — 3W,  i8»,  and  —  iiw  ; 

5a*:x:,  —  3a*^,  a^:x;,  and  —  3a^,T  ; 

3w/>*,  —8mp',  sa'x,  —^mp',  —^a'x,  and  2a:x;'  ; 

230%  5b',  —Sa^b',  —136%  24a'6%  and  —19a'  ; 

a'bx,  i2a^bx,  and  —ga^x ; 

-iS(a5(;='+3),  27(a:x;"+3),  and  -9  (ax'+s); 

13(^-3'),  -«(^-:y),  4»»(^->'); 

mxy  and  +«^;y  ; 

c:x;2,  {f^2^  —ex',  —dx'  ; 

Sa+  36+9C 

2a  — 106  — 2c  ; 

6a  +  i5c  — 176— 8(i  and  —70  +  12(^  +  156  — 12c  ; 

'ja'  +  2^b'  —  i$c'  and  — 216'— 6a'  +  i2c^  ; 

2gxy—2'jy'  +  2/^x'  and  —  145c:)'— 36^2+36^'  ; 

— 3a— 76  +  14C  and  —  na  +  206— 34c  ; 

i4ife— 11/+12W,  —3^  +  12/— 6w,  and  — 12^+/  — 2w; 

—  i8a*6^  +  i2a4-8a36  and  4^063 -02634.3103^. 


Fundamental  Processes  Applied  to  Integral  Expressions    99 

§  46.     Subtraction  of  Integral  Algebraic  Expressions 

If  two  numbers,  as  2  and  5,  are  given,  then  7  can  be  found 
from  2  and  5  by  addition.  On  the  other  hand,  if  7  and  5  are 
given,  then  2  can  be  found  as  that  number  which  added  to  5 
gives  7.  If  2  and  7  are  given,  5  can  be  found  as  that  number 
which  added  to  2  gives  7. 

Definition. — To  subtract  the  number  b  from  the  num- 
ber a  means  to  find  that  number  which  added  to  b  produces 
a;  i.  e.,  to  find  the  number  of  units  which  are  to  be  counted 
from  b  to  get  to  a. 

To  indicate  that  b  is  to  be  subtracted  from  a  the  symbol  (— ) 
is  written  between  b  and  a,  as  a—b,  which  is  read  "a  minus  b." 

The  process  of  finding  a  number  which,  added  to  a  given 
number,  produces  a  second  given  number  is  called  subtraction. 

I.  According  to  the  definition  of  subtraction  find  the  values 
of  the  following  differences: 

(i)  9-5; 

(2)  6a  — 2a; 

(3)  8°-6°; 

(4)  What  number  added  to  —6  gives  a  ? 
Find  the  following  differences  : 

(5)  d-i-b);  (8)  9^-(-3«); 

(6)  i6-(-3);  (9)  iox-{-sx); 

(7)  2i-(-i5);  (10)  25-(-9)-(-6); 

(11)  xoy-{-zy)-{-^y); 

(12)  What  number  added  to  +6  gives  —a? 

(13)  Find -a -(+6);  (16)   -i8a-(-|-i6a); 

(14)  Find  -s-(-3);        (17)  -2iy-{+2(>y); 

(15)  -l^b-{  +  lsby,  (18)   -6c6-(  +  3c6); 

(19)  What  number  added  to  —b  gives  —a? 

(20)  Find  —a  —  {—b);         (24)   —175  — (—45); 

(21)  -^^b-{-^b)^,  (25)   -8m-(-8m); 

(22)  -7^_(_7^);  (26)   -Ud-{-cd); 

(23)  -i29-(-5?);  (27)   -iizl-{-i6zl)  . 


loo  First-Year  Mathematics 

Make  a  rule  for  the  subtraction  of  positive  or  negative 
numbers. 

2.  According  to  this  rule,  subtract  the  lower  number  of 
each  of  the  following  pairs  from  the  upper  number: 


13       -7 

-  5           -8 

7a            idhx^ 
4a          —;^bx^ 

-75 
+  25 

-i8tn' 

5  (a -2*3) 
1 1  (a  — 2*3) 

+  9  12 
+  21         -  5 

—  ^x'V         - 

X             4  +  M 
-5                 -2 

26'y3-yS           IK^CX^ 

■  yv^ys           3cic» 

6m^px 
—4m'px       — 

iSnt'(x-y) 
-27,m'{x-y) 

—  2x(i -\-sa'y) 
-i4x(i +sa'y) 

3.  Add    /»3  4-3^2-1-4^  —  6,     —p'  —  2p  +  i,    p^  —  i,   and 
3^3  -I- 2/> -I- 2  . 

4.  Add  x^-\-2xy-{-y^,x'—2xy+y',x^—4xy,  and  4:x;3'4->'*. 

5.  Add    —  23a»6  +  4ia3c  +  56^*6  — 156"^,     —  6a*6  +  26a"c 
+  59c^6  — 266'c,    25a^c  — I9&*c  +  i8c^&. 

6.  Add    ix'  —  \xy—\y^,    —  x' —  ^xy -{■  2y^ ,    and    ^x'  —  xy 

7.  A  conductor  collects  during  a  trip  11  dollars,  20  quarters, 
40  dimes,  62  nickels,  and  18  cents.  He  pays  out  2  dollars, 
24  quarters,  15  dimes,  10  nickels,  and  6  cents.  How  much 
does  he  make  on  the  trip  ? 

8.  From  a'  +  2ab-\-b'  subtract  a'  —  2ab-{-b\ 

9.  Fromx^+^x'y+^xy'  +  y3  suhtrsLCt  —sx'y-hsxy'—^y^. 

10.  From   a^b' +  a*+b* —  ^a^b  +  ^ab^     subtract    —a'^  —  b* 
—a'b'—^a^b—^ab^. 

11.  From  c^  —  2a'c+d^  —  r'  subtract  c^—a'c—r'. 

12.  What  is  the  meaning  of  a  —  (b  —  {c+d))  ? 

13.  How  would  you  remove  the  parentheses  in  problem  12  ? 
In  examples  14-17  remove  the  parentheses,  without  chang- 
ing the  values  of  the  expressions  : 

14.  9x-\sy-{6y+7z)-i7y-4z)\. 


Fundamental  Processes  Applied  to  Integral  Expressions  loi 

15.  29a -[15c-  { 14a +  236- (17^  +  56 -38^)  -(13C+  186 

-230)  (]• 

16.  ;^a''x—[2mb+\a^x—(—4s't-\-$mb)+s''t\  ]. 

17.  -{-{-(x-y))). 

§  47.     Multiplication  of  Integral  Algebraic  Expressions 

If  in  the  sum  the  addends  are  all  equal,  the  symbol,  X, 
or  the  dot,  (•),  is  written  between  one  of  them  and  their  num- 
ber.    Thus    a+a+a=;^Xa    or    3  •  a,    b  +  b+b (to    a 

terms)  =a  -  b. 

This  method  of  forming  out  of  two  numbers  as  a  and  b  a 
new  number  is  called  multiplication.  The  result  of  the  multi- 
plication is  called  the  product,  and  the  numbers  which  are 
multipUed  are  called  the  factors. 

If  the  numbers  are  represented  by  letters  the  multiplication 
symbol  may  be  omitted.  Therefore,  aXb,  a  •  b,  and  ab  have 
the  same  meaning. 

Do  2X3,  2  •  3,  and  23  have  the  same  meaning?  May 
the  multiplication  sign  between  two  arithmetic  numbers  be 
omitted  ?  The  multiplier  i  may  or  may  not  be  omitted. 
Thus  I  •  a=a. 

When  the  multiplier  i  is  omitted  it  is  said  to  be  understood. 

What  is  the  meaning  of  5-2?  ^'3?  4*6?  Show  that  as 
2  .  3=3  •  2,  so  a  •  b=b  •  a. 

Illustrate,  by  taking  any  particular  set  of  figures,  that 
abc=acb=cab=cba=b  ca. 

Prove  the  equahty  of  the  same  products,  assuming  that 
ab=ba. 

State  the  general  principle  just  illustrated. 

Multiply  as  indicated : 

1.  4X13X15.  5-  6aX7Xb. 

2.  7X8X25.  6.  7(aXbXcXd). 
3.4(2X3X6).  7.  jaXjbXycXyd. 
4.  5^X4.  8.  axXbxXcx. 


I02  First-Year  Mathematics 

Point  out  the  difference  between  (6)  and  (7). 

9.  ^aX^by.2<|X .  13.  abXbcXac . 

10.  ^azXybdzXScXsy  '  '  i4-  a^Xa' . 

11.  2abzX2,bczXAacy .  15.  fe'fts . 

12.  abXab  .  16.  rv^X^^. 

17.  Show  that  a'*Xa**=a*"+*',  w  and  «  denoting  whole 
numbers.  Formulate  into  words  the  law  of  exponents  in  mul- 
tiplication expressed  by  this  equation. 


Multiply  as  indicated: 

18.    a»63c4a4j4c3  . 

27. 

8aw3:x;'X5a>'3. 

19.  26X3'. 

28. 

m^x^XzP^xy^ . 

20.  e,x'y^  X  7  X  y^z^  X  2xH  . 

29. 

^a^m'x"*  ^2bm^ 
X 
4              9 

21.  13+33  +  53. 

22.  i*  +  7=»  +  io*. 

30- 

aX-b. 

23.  6=»+8^. 

31- 

2W/3X  —  3W/'. 

24.  43  XS^ 

32. 

a='^3X-a3:v4. 

25.  6='+8=». 

33- 

amX  —am'x. 

26.  2mH'X2,mt2 . 

34- 

2pq'X-3P'q*' 

35.  Write  in  the  form  of  squares,  cubes,  and  other  powers 
the  following  expressions:  gp',  a'x',  4m^,  i2i3(;*,  492*,  Siiic^^ 
a'b'x'y',  i2ix^y^,  36af6^  a'b'^c^,  Agx'y^,  Sx^,  x^y^z^,  343^3, 
2'j(x—y)3,  'j2g(x+i)^,  64{a^  +  by,  ;i2x^,  'j2gx^,  24;3x5ySzs, 
2i6a^,  2$6m^n^^,  128a'. 

To  indicate  that  a  sum,  as  b+c,  is  to  be  multiplied  by  a 
number  a,  the  sum  is  inclosed  in  a  parenthesis,  thus  a(b+c). 

Since  a{b+c)  =  (b+c)  +  (b+c)+ (to  a  terms) 

=b+b-\-b  +  b (to    a    terms)+c+c+c 

(to  a  terms) 

=ab+ac . 

The  product  of  a(b-{-c)  may  be  obtained  by  multiplying 
each  term  of  the  multipHcand  by  a  and  then  adding  the  products. 
Perform  the  indicated  operations: 


Fundamental  Processes  Applied  to  Integral  Expressions  103 

1.  a(h—c) .  9.  a  +  hXc. 

2.  {h+c)a.  10.  {x-]ry)z-^k. 

3-  3  +  (4  +  2).  II.  J+3(^+«)- 

4.  3+4X2.  12.  8xj(8^— 9)'). 

5.  a+&X6.  13.  i6x3'z(23(;y— 4;>'z+65c:2)  . 

6.  3  +  (4X2).  14.  a  +  (&+c):x;. 
7.3+4X2  +  7.  1$.  ax-]-hy-\-{a—h)y. 

8.  (3+4)2  +  7.  16.  5a(6a  — 75)+86(9a  — 106) . 

17.  i6z[(29Z— a) +  2(80  —  142) +  76]+4z(286— 3a). 

18.  5a2(5  +  6a='-7a4), 

19.  i7a='(2a2— 36^  — sa^c) . 

20.  7w^w3(2w— 3W— 4w»+6m3») . 

21.  2ax{^x^—^a^x^-\-$ax^). 
Problems  and  exercises  : 

1.  Formulate  into  words  the  following  equations: 

a{h-\-c)=ab+ac;  ab-\-ac=a{b+c)  ; 

a(b—c)=ab—ac;  ab—ac=a(b—c). 

2.  Show  similarly  how  to  find  the  product  of  a  monomial 
by  any  polynomial. 

3.  State  whether  these  equations  are  true: 

(i)  a(b+c—d—}r)=ab+ac—ad—afr? 
(2)  ma+mb—mc—md='m(a  +  b—c—d)? 
Give  reasons. 

4.  Write  in  equivalent  factored  forms  the  following  ex- 
pressions: 

(i)  5a  — 106  ;  (4)  i$x^  —  iox^  +  $x'; 

(2)  i7:x;^  — 289:^:3  J  (^)  3^5  — i2w3»+6w«'*  J 

(3)  i6x'  —  2abx;  (6)  I4x'y'z'  —  'jx3y3z'+8xy'z'', 

(7)  tom^nH^  —^^m^n^r^  +gom^n^r3+gom^n3r' . 

5.  Show  that  according  to  the  definition  of  multipHcation 
(+a)  X(+b)  =  +ab  and  (+a)X(-b)  =  -ab  {a  and  b  being 
positive  integers).  Why  have  (— a)X(  +  6)  and  (—a)X(—b) 
no  meaning  according  to  the  same  definition  ? 


to4  First-Year  Mathematics 

Definition.  The  last  two  cases  can  be  included  by  en- 
larging the  definition  of  multiplication :  To  multiply  one  alge- 
braic number  by  another  means  to  find  the  product  of  the 
multiplicand  by  the  absolute  value  (value  disregarding  the 
sign)  of  the  multiplier  and  then  to  prefix  to  this  product  the 
sign  of  the  multiplier. 

From  this  definition  it  follows  that: 

(i)  (-F-a) X (+&)  =  + (+06)  =  -Fa6; 

(2)  (-ha)X(-6)  =  +  (-a6)  =  -a6; 

(3)  {-a)X{+b)  =  -{+ab)  =  -ah- 

(4)  {-a)X{-h)  =  -{-ab)  =  -\-ah. 

Thus  it  appears  that  in  multiplication  numbers  having  like 
signs  give  positive  products;  and  numbers  having  unlike  signs 
give  negative  products. 
Exercises  : 

6.  Simplify: 

(i)  a-7(6-hc);  (3)   -a'X{-aY; 

{2)  x—(){y—2)  ;  (4)   — oX— a^X— a3; 

(5)  -aX(-a)-X(-a)3; 

(6)  (pqr)Xi-p'q'r-)X(p'q'r^); 

(7)  i4a-l-2b—2(^a—4b)  +  {a+b)  ; 

(8)  $a(6a  —  'jb)—Sb(ga  —  iob)  ; 

(9)  (25a^  — I3a-Hii6)2a  — (7a  — i56)4a  ; 

(10)  [3^-7(5^+301-0(3^-6^)) 4-^(35^-60) ; 

(11)  'jxy^-[{i8a  +  sb)x+sb{y-x)]z-2y(sb-'j6)  ; 

(12)  28a-5J3[7a-f5(56-3a)-|-i6(5a-3&)]-2i(6-a)f. 

7.  Factor  the  following  expressions: 

(i)  4^+4*;  (5)  (a+b)x-(a  +  b)y; 

(2)  ax+bx  ;  (6)  i6a'x—24ax'  —  i4a'  +  2iax; 

(3)  ax+ay—az  ;  (7)  ax+ay+bx+by  ; 

(4)  a(c+d)—b(c+d);  (8)  ac—ad+bc—bd; 

(9)  ^6a'xy-\-48ax'y—i6axy'  —  'jabc—6xbc  +  2ybc. 


Fundamental  Processes  Applied  to  Integral  Expressions  105 

8.     Find  the  value  of 

(i)  67  •  7-23  .7  +  7  .35; 

(2)  12  •  157  —  72+4  •  12  —  12  •  40. 

Since  a'>io=a(b—b)=ab—ab=o;   .'.  aXo=o. 

Conversely,  it  is  evident  that  a  product  is  o  if,  and  only 
if,  at  least  one  of  its  factors  is  o.  Hence,  (x—^)(x—8)=o  if, 
and  only  if,  :x;— 5=0  or  ic— 8=0;  or  if,  and  only  if,  x=^  or 
x=S. 

From  this  example  it  is  seen  that,  if  in  an  equation  all  the 
terms  be  removed  to  one  side,  factoring  that  side  may  enable 
us  to  find  the  value  of  the  unknown  in  the  given  equation. 

The  values  of  the  unknown  in  an  equation  are  called  the 
roots  of  the  equation. 

Exercises  and  problems: 
I.  Solve  i2z^=4Z.  2.  Solve  5:^;*  — 7:x;=o. 

3.  Solve  (x—i)(x+i)(x—2)=o. 

4.  The  square  of  a  number  less  five  times  the  number 
equals  zero.     Find  the  number, 

5.  The  square  of  a  number  added  to  twice  that  number 
less  four  times  this  second  number  is  zero.  What  is  the 
number  ? 

6.  The  sum  of  a  square  and  a  rectangle,  one  of  whose 
sides  equals  the  side  of  the  square,  and  the  other  side  2 
inches  less,  equals  the  area  of  a  rectangle  with  one  side  equal 
to  the  side  of  the  given  square  and  the  other  5  inches  greater. 

7.  Solve  a  x+bx+cx=ac+bc+c^. 

8.  Multiply: 

(i)   {a  +  2b)(3a+4b)  ; 

(2)  (-Sx+sy){-2x+sy) ; 

(3)  (y+mh-¥) ; 

(4)  (24^m-^^m  +  5lp)(i2m+24n-s6p)  ; 

(5)  5lx+32y-4z  +  7^u){iox-2oy+4z-Su)  ; 

(6)  {6x^  +  2x+i)(x'—x+i)  ; 


io6 


First-Year  Mathematics 


(7)  {x^-\-^xy-\-y^){x^+xy-y')  ; 

(8)  {x+y){x^-\-T,x^y+7,xy^  +  y3)  ; 

(9)  {a^-^ah-b'){a^-\-2a^h-\-ah^-^h3)  ; 

(10)  (p'  +  2pq+q^)(p'-2pq  +  q')  ; 

(11)  (x3+2x'y+4xy'  +  syi)  (x-y) ; 

(12)  (^a  +  2b)(2ax—a'—x')(bx—2a)  ; 

(13)  (3i'-5^-^2s)(s-2t+v)(s-s-t) . 

9.  Show  that  the  following  five  equations  are  true: 

(i)  (a  +  by=a^  +  2ab+b''  ; 

(2)  (a-b)'=a'-2ab-\-b'  ; 

(3)  (a+b)(a-b)=^a'-b-; 

(4)  (a+by^a^+^a'b+sab'+bi  ; 

(5)  (o-&)^=a3-3a*&+3a&^-363. 

10.  The  length  of  the  room  exceeds  its  breadth  by  5  ft. 
If  the  length  had  been  increased  by  3  ft.  and  the  breadth 
diminished  by  2  ft.  the  area  would  not  have  been  altered. 
Find  the  dimensions  of  the  room. 

§  48.     Geometrical  Representation  of  Special  Forms  of  Products 

I.  Show  from  the  figure  that  xy+xz=x(y-\-z). 


A 

X 

K 

1 

X 

Fig.  57 

2.  Show  from  Fig.  58 

that 

ac- 

-ad+bc-\-bd={a-\-b){c-\-d). 

<\, 

I 

A 
1 

d 

a 

I 

4. 

Vi 

c 

^ 

ir 

\ 

c 

0.-4.  X--  - 

V 

Fi 

0.58 

Fundamental  Processes  Applied  to  Integral  Expressions  107 


3,  Show  from  Fig.  59  that 

a'  +  2ab  +  b'  =  (a+by. 

4.  Show  from  Fig.  60  that 

a'-b'  =  (a+b)(a-b). 


a. 
0.1 

* 

a* 

al 

\ 


<- 

a. > 

I 

<K-^ 

1 

I 

L--^ 


Fig.  60 


T 
Fig.  59 

5.  Show  by  the  aid  of  a  divided  rectangle  that 

{a-\rb){c—d)=ac—ad-\-bc—bd. 

6.  Similarly  work  out  the  value  of  (a—b)(c—d)  with  the 
aid  of  a  divided  rectangle. 

7.  What  does   the   equation   (+a—b)(+c—d)=ac—ad  — 
bc+bd  become,  when 

(i)  6=0,     and  d=o? 

(2)  b=o,       '•     c=o? 

(3)  0=0,       "     c=o? 

8.  Compare  each  of  the  results  of  (7)  with  the  expressions 
(i),  (2),  (3),  and  (4),  p.  104,  under  problem  5. 

9.  Multiply  by  inspection: 
(i)  (x+yy;  (6)  (x+3y 

(2)  (m+ny;  (7)  (a-sY 

(3)  (h+ky;  (8)  (7-yy 

(4)  (a-py;  (9)  (9a-7by 

(5)  (c-^)';  (10)  (2^-3:v)= 


(11)  (4a  +  7xy; 

(12)  {T,m*  —  2ny; 

(13)  (2^^-3)»; 

(14)  (2a^i»?+36:v3)= 

(15)  [(a+6)+c]- 


xo8 


First-Year  Mathematics 


(i6)  [(a+6)-4cp;  (20)  (2a+36)3; 

(17)  (a+6)»  +  (a-6)-  (21)  (3a-4^')^; 

(18)  {^x+2>yY-{2x-yY;  (22)  (4^^6  +  506^)3. 

(19)  (3a+86)^  +  (4a+66)^-(5a-io6)^; 

10.  Transform  the  following  differences  into  products: 


(i)  x^-y'; 

(2)  36^=^-253'^ 

(3)  i6a'-gb'; 


Fig.  61 


(4)  iooa4-8i6^;  (7)  67^-33^; 

(5)  i44J(;='3'^-8ia2>'^;         (8)  81^-19^; 

(6)  97^-3";  (9)  1017^-17^ 

11.  A  and  B  are  stakes  on  opposite 
sides  of  a  lake.  AC  =  575  rd.,  60=425 
rd.,  angle  ABC =90°,  therefore  AB^ 
=AC='-BC^  What  is  the  distance 
across  the  lake  ? 

12.  The  difference  of  the  squares 
of  two  consecutive  numbers  is  25.  Find 
the  numbers. 

13.  Solve  for  :x;:   (x+b)^—c'=o. 


§  49.     Division  of  Integral  Algebraic  Expressions 

To  divide  a  number  by  a  second  number  means  to  find  a 
third  number  (quotient)  which,  when  multiplied  by  the  second 
(divisor),  gives  the  first  (dividend).  According  to  this  defi- 
nition: QuotientXDivisor= Dividend;  i.  e.,  Division  is  the 
reverse  0}  multiplication. 

To  indicate  that  a  is  to  be  divided  by  b,  the  symbols  :,-;-, 

/,  or  -,  are  written  between  a  and  b;    thus,  a-^b,  a/b,  and  -  . 

I.  According  to  the  definition  find  the  quotient  in  24W4-4; 
ab/a;  3006/5;  a^/a;  a^/a^;  x^y'^/xy';  3905/13^;  4Sx3y5/i2xy'] 
(ax+bx)/x;  {ax— ay)/ a;  (7  +  7a)/7;  (ab+a)/a;  14/3;  120/56. 

If  an  integer  cannot  be  found  which  multiplied  by  6  pro- 
duces o,  then,  as  in  arithmetic,  the  indicated  division,  -,  is 
called  a  fraction. 


Fundamental  Processes  Applied  to  Integral  Expressions  109 

Since  aXo=o,  it  follows  from  the  definition  of  division 
that  o/a=o. 

2.  What  is  the  value  of  0/0?  (Use  definition  of  division 
and  the  equation  aXo=o). 

Since 
(+a)X(+b)  =  +  (a'b),      therefore  +(a'b)~(+b)=(+a)  ; 
i-a)Xi+b)  =  -(a-b),  "        -(a-6)^(+&)  =  (-a); 

(+a)X(-b)=^-{a-b),  "        -(a.6)-(-6)  =  (+a); 

(-a)X{-b)  =  +  {a'b),  "        +(a-b)^{-b)  =  (-a). 

These  results  may  be  formulated  in  words  thus:  In  divi- 
sion, numbers,  having  like  signs  give  positive  quotients,  and 
numbers  having  unlike  signs  give  negative  quotients. 

3.  Find  the  quotients: 

+206  —  i6x  —63^^  —35^^ 


-5a  '            2y     '         -  7^^  ' 

-IP'  ' 

What  is  the  quotient  of 

0             s   ^         X     ^         p 

¥               S4                 x^                 p^ 

X-"  ' 

Can  the  quotient  be  obtained  in 

this  way  with  the 

ing: 

r7  y-" 

6.  How  may  the  exponent  of  the  quotient  in  each  of  the 
parts  of  Problem  4  be  obtained  from  the  exponent  in  the 
dividend  and  divisor  ? 

7.  Formulate  into  words  the  law  of  exponents  in  division; 

viz. : 

a'" 

a» 
In  the  problems  from  which  this  law  is  obtained  the  exponent 
in  the  dividend  is  greater  than  in  the  divisor,  i.  e.,  m>n. 

We  will  now  consider  cases  where  m=n,  and  where 
m<n. 


no  First-Year  Mathematics 

§ 50.     Cases  in  which  m=n,  and  m<n. 

1.  Case  m=n. 

Suppose  that  the  law  is  true,  then  -^  =  7""*  =  7°. 

According  to  the  definition  of  division  —  =  i .     Of  the  two 

answers  obtained  the  first  one,  7",  has  no  meaning  according 
to  our  present  idea  of  the  meaning  of  exponents;  for  we  can- 
not take  7  zero  times  as  a  factor.     If  we  agree  that  7°  shall 

mean    i,   then    the    exponential  law  gives  —  =  7* "■^  =  7°  =  !, 

a  result  which  agrees  with  that  obtained  from  the  definition 

of  division. 

In  general,  a°  is  defined  by  the  equation  a°  =  i.     Then 

a"* 
the  exponential  law  gives  for  w=»,  — =a"*~'"=a°  =  i,  which 

conforms  to  the  definition  of  division. 

2.  Case  m<.n,  or  n=m-{-k. 

Suppose  that  the  exponential  law  holds.     Then 

a"" 


_^m-(m+fe)_^-* 


Just  as  before,  so  here,  the  result  has  no  meaning  accord- 
ing to  the  definition  of  exponent.  Show  why  it  has  not.  It 
will  be  found  advantageous  to  agree  that  a~*  be  interpreted 

to  mean  -r,  1.  e.,  a~*=— 7  . 

Then  both  the  exponential  law  and  division  lead  to  the 
same  result.  For,  assuming  that,  as  in  arithmetic,  dividend 
and  divisor  may  be  divided  by  the  same  factor,  then 


With  these  two  new  definitions,  a°  =  x,  and  a~^=—r,    the 

a" 


Fundamental  Processes  Applied  to  Integral  Expressions  iii 

exponential  law  holds  for  any  positive  integral  numbers,  m 
and  n. 

Write  the  eqmvalents  of  the  following  expressions  without 
using  exponents : 

2-3,    3-%   a)-s   (*)-%    p-3. 

§51.     Exercises 

Reduce  the  following  to  lowest  terms  by  dividing  out  all 
common  factors  in  numerators  and  denominators: 
378  —2^a''bc'  ax— ay 

■  —63'  ^"      5as6c     '  ^'      —a 

2.^—^.  6.-^ ; .  10.  — . 

—  DC  —i^ab^mc'  m 

—  dx'^y  —a^b^c"^  as-^bs—cs-\-ds 
3-  —t:;^,-              7 

4 


3xy  ''     a^b^'c^  '  s 

8:v*y^  ax-\-bx  T,ar+^ab 


X  ^axy 

§  52.     Division  of  a  Polynomial  by  a  Monomial 

State  a  short  way  of  dividing  a  polynomial  by  a  monomial. 
Reduce  the  following  to  their  lowest  terms: 
2oa3b—  1 5a^6^  4-30063 


(i) 
(2) 
(3) 
(4) 
(S) 
(6) 


sab 
lox'y—  i^x^y^  +  ^x^y 

-Sxy 
^6x^y'*—42xSy^z 

—  6x^y 
35a6%3  — 42a3&4c3 

—  jab^c 
2(a+b)5-s(a-\-b)3  +  2(a+by 

(a  +  by 
2(tn—xy  —  2(m—x)3  —  6(m—xy 
(m—xy 


112  First- Year  Mathematics 

X    §  53-     Division  of  a  Polynomial  by  a  Polynomial 

Multiply  {x^  —  'jx'-\-2x-\-4)  by  (45c;* +  2:^—3).  The  com- 
plete work  consists  of  four  steps,  thus : 

(i)  43(;*X(:x;3  — 7^2  +  2:x;+4)=4xS  — 28:x;4+8:x;3  +  i6:x:='  ; 

(2)  2xX(xi  —  'jx'  +  2X-\-4)  =  2x'^  —  i4X^-\-4X'  +  8x  ; 

(3)  —3X(x3  —  'jx^-{-2x-\-4)  =  —2X^-\-2ix'  —  6x—i2  ; 

.'.  (4)  (4X'-\-2x—;})(x3  —  'jx^  +  2x+4)=4x^  —  26x^—gx^-\-4ix' 
-l-2:x;— 12  . 

Suppose  next,  that  we  have  given  for  the  dividend,  the 
right  member  of  equation  (4),  and  the  second  factor  of  the 
left  number  as  the  divisor.  The  problem  is  to  find  the 
terms  4X',  2x,  and  —3  of  the  quotient. 

It  will  be  convenient  to  arrange  dividend  and  divisor  ac- 
cording to  powers  of  the  same  letter,  before  beginning  to  divide. 
See  the  arrangement  here  given. 

4:x;S  — 26^:4—  9x3-}-4i3£;^-}-25£;— 12  I  x3  —  'jx'  +  2x-]-4 

4^5-28^4+  8^3  +  i6:x:^  4:^^  +  2^-3 


2X*  —  1 7^3  -\-2^X'  -\-2X 

2x^  —  14x3+  4:!<;2-l-8:x; 

—  3x3-|-2i:x:^  — 6:);— 12 

—  35[;3-f  2i:x;*— 6:x;— 12 

Explanation. — First  we  are  to  find,  4X'',  the  first  term 
of  the  quotient.  4X^  was  found  by  multiplying  4:3c*  by  :x:3. 
Therefore  4X^  is  found  by  dividing  4:^5  by  ;x;3. 

Therefore,  the  first  term  0}  the  quotient  is  obtained  by  divid- 
ing the  first  term  0}  the  dividend  by  the  first  term  of  the  divisor. 

Next,  to  find  2X,  the  second  term  of  the  quotient. 

Since  4X'  is  now  known  we  can  form  equation  (i).  Sub- 
tracting its  right  side  from  the  dividend,  the  first  term  of  the 
remainder  is  2x*.  2X*  was  found  from  2:x:X^^.  Hence  2x^-t-x^ 
gives  the  second  term  of  the  quotient.  Therefore  the  second 
term  of  the  quotient  is  found  by  (i)  multiplying  the  first  term 


Fundamental  Processes  Applied  to  Integral  Expressions  113 


of  the  quotient  by  the  divisor;  (2)  subtracting  this  from  the 
dividend;  and  (3)  dividing  the  first  term  of  the  remainder 
by  the  first  term  of  the  divisor. 

Similarly,  it  can  be  shown  that  any  term  of  the  quotient 
is  found  by,  (i)  forming  the  product  of  the  preceding  term 
by  the  divisor;  (2)  subtracting  this  from  the  last  remainder; 
and  (3)  dividing  the  first  term  of  the  new  remainder  by  the 
first  term  of  the  divisor. 

Find  the  quotient  of  the  first  polynomial  divided  by  the 
second,  in  each  of  the  following  problems: 
(x^  —  2xy+y^)-i-(x—y). 
{a3+^a'b+s<^b''+b3)^(a+b). 
(6a^+;^iab+^^b^  +  i2ac-\-42bc)-^(2a  +  'jb). 
{x'  +  'JX+ 12)  -7-  (x+ ;^) . 
(x^-i^x+so)^(x-s). 
-f^+i)H-(x-f). 
-i8|:x;+i2)-f-(:x;— I). 
(3a4+3a2  +  5+3a+3oS4-5a3)-=-(i+a). 
{24m^+42mn  —  62mp+i^n*  —  46np-\-2^p')-i-(6m  +  3» 

-5P)- 

{a3+b3+c3-^abc)^(a-\-b+c). 

{2a4+k'^-sa3k-4ak3+6a'k')^(k'+a'-ak). 

(a4-i664)-^(a-26). 

{x^  +  y5)-h(x+y). 

(x^  +  y3+z^  —^xyz)  -r-  (x'  +  y'  +z*  —xy— yz—zx). 

{6px^  +  (914$  ~  9P)xy  +  i6vxz  +  louvx—  24ryz  —  2iqy 

—  i^uvy)-i-(^px-{-'jqy-\-Svz  +  ^uv). 


I. 
2. 

3- 
4. 

5- 
6. 

7- 


10. 

II. 
12. 

13- 
14. 

15- 


CHAPTER  XVI 

FRACTIONS 

§  54.     Fractions  Having  the  Same  Denominator 

It  was  shown  previously  (p.  26,  Ex.  6,  (12),  (13)  that 

a-{-b_a    b 

c        c     c 

,  a—b    a    b 

and  = . 

c       c     c 

By  interchanging  the  sides  of  each  of  these  two  equations, 

we  get: 

a    b _a+b 


and 
Similarly, 


c  c  c 
a  b_a— 
c     c       c 


a      &      c      d     a+b+c+d 


m    in    m    m  m 

Translate  this  last  equation  into  words  and  obtain  a  rule 
for  addition  and  subtraction  of  fractions  that  have  a  common 
denominator. 

Combine  each  of  the  following  expressions  into  a  single 
fraction. 

I-  M+2^7+if  •  oc-y_^2y 

X    y  „    c—ax  ,  c  +  'zax 

3.  — ^.  8.  +  — ^— . 

22  45  45 

X    y  3a    30—86 

a    a'  5b        $b      ' 

a     b     c  io:v    i2x      lix 

5. .  10.  +  — — -^  . 

XXX  lya     17a     17a 

,    a+b  ,  a—b  „,         .    ^      a        —a      J^a 

6. 1 .  II.  Show  that —7= +-7-  = -1 r- 

22  0  b  —b 

114 


Fractions  115 

§  55.     Fractions  Having  Diiferent  Denominators 

To  show  how  to  add  or  subtract  fractions  having  dij6ferent 

(t     c 
denominators,  take,  for  example,  7  +  ;^  and  multiply  it  by  hd, 

thus: 


0+0^^=^ 


0+3)*^- 


Xhd+^-Xhd'     Why? 
=ad+hc.  Why? 

d+hc.  Why? 


a    c_^ad+U 

b^d        bd  ^ 

1.  In  a  similar  way  show  that  7—^= — ri —  • 

0    a        bd 

_        .        a     c     ad±:bc  .  ,  1     ,      .  , 

2.  Translate  t±3= j — mto  words,  and  obtam  a  rule 

b    a        aa 

for  addition  and  subtraction  of  fractions  having  different  de- 
nominators. 

3.  According  to  this  rule  show  that  we  must  have 
34     _3(rv='  +  i)+4(^-i)_  2,x^+^x-i 


x—i     x^-\-i       {x—i){x^  +  x)  x^  —  x^  —  x—i 

4.  Express  the  following  sums  and  differences  as  a  single 
fraction : 

(i)  i+i  ;  (6)^^"^^    2x-a  ^ 


(2) 

^      X    y 

(4)  ^+'^= 


(7) 
(8) 


x—2a 

x—a 

> 

I 

I 

x+y 

x—y  ' 

X 

X 

l-X' 

I+X'   ' 

I 

^  , 

I 

(5) 


(9) 
a-\-x    a~x  ^  x—i     x+i     x—2 

a—x    a+x ' 


ii6  First-Year  Mathematics 

In  case  the  denominators  of  the  fractions  to  be  added  have 
a  common  factor  the  fractions  may  first  be  reduced  to  equal 
fractions  having  for  their  denominators  the  least  common 
multiple  of  all  the  given  denominators,  and  then  be  added 
and  simpKfied. 

For  example: 

5.  Combine  the  following,  each  into  a  single  fraction: 


2     5     10  '  a+2b    ;^ad+6bd  * 

302  X 

(■i)  ^— ^  ;  (Suggestion. — Put  6=-.) 

^^^  12     20  '  ^  I   ' 

.ah  /    \  ^ 

(4)  — +~  ;  (15)  — a  ; 
ex     cy  -"  y         ' 

(6) ;  (17)  \T.x-!-  ; 

c     ca  3 

.      a—h    b—c    c—a  .  a        b 


^^ — \-— — T~  ;  (20) 1 

xy    xy^       x'y^  x        ;ix 


,    .      a  ab  ,    .  a— 2b    4a— <6 

(10) 7 r  ;  (21) ^ ^ 

a— I     a(a— 1)  ^x  ^x 

(12)    _^-+-i^±>^    •  (2S)     -  +  -  +  -  • 

2X—2y    4x'—6xy  '  6c    ac    ab ' 


Fractions  117 

(24) ; 

x—y    y 

^^5)  — I 1 — + — I —  ; 

/  ..  5J;-4y+3Z  I  2x+3>'-4Z     ^^x-dy-^z 
(26) 1 ; 

X  3J  2Z 

,    .     X       y       z  {a-\-hy 

(27)^-^+^;  (29)  -^^-i; 

(28)    ^-^   ;  (30)    -^+^^-^'  . 

^    '      a       a-b  '  ^"^  ^  ic^-i     5c;-i     ic+i 

6.  Solve  the  following  equations: 

,  .  X    X  X  r  \  y-^  I 

(3)  ^^7_^_3^z:4^^  (^)  T^^-^=-^-i  . 

^"^^      12        2     4X-3        '  4^     SJ 

7.  The  denominator  of  a  fraction  exceeds  its  numerator 
by  2 ;  and  if  i  be  added  to  both,  the  resulting  fraction  will  be 
be  equal  to  §.    What  is  the  fraction  ? 

§  56.     Multiplication  of  Fractions 

How  are  fractions  multiplied  in  arithmetic?  Fractions 
in  algebra  are  multiplied  in  the  same  way  as  in  arithmetic 
thus: 

y+x      x'—y'       m+n    6(m'—n') 
tn+n     i2(m+n)     m—n         x+y 
2 

_  (jc' — y')(m' — n')  _  x'  —y' 
2(m+n)(m—n)  2 


Its  First-Year  Mathematics 

EXERCISES 

Perform  the  indicated  operations: 

I.  |-f -i-  a*b^     6x*y' 

14. 


2-  *•*•  4x'y^    5<i^b' 

3.  --ac.  15. 


a-6    a='+6'' 

4.—-a3.  16.  — ^; -^. 

a'  Sy  +  Sx      3 

a    I                                              ah 
1;.  -  •  — .  17.  • . 

^    h    X  '    a+b    a—b 


i2ab+Scd  '      ^xy'      8(x—y)' 

a    c  a'—ab    x'+xy 

7.  -•-, .  10. f. 

'    c    d  ^    x'—xy    a'+ab 

yxyz       ,  a3— 863         a +  26 

8.  ;    •  oabc .  20.  7-  • ; — r- 

^bc    ^  a''-4b'    a^  +  2ab+b' 

ah    yz  g  +  i    a  +  2    a  +  <, 

xy    he'  '  a— I    a— 2    a— 3' 

2db    WJC  :x;2— :x:r    (a+6)^ 

10. TT--  22.    r-T'T V," 

33(;;y    66>'  3<J^+30    (^— y) 

ibxy    2<bc'  \x    y/\x    y/' 


II. 


i6xy    2$bc ' 

^ab    5&C    jxz 
4xy    6yz    Sac 


36'y    7»^"    4a  V  y/\a+x/ 


'    V^     a       /  \a'     a      / 


Fractions  IJ9 

§  57.     Division  of  Fractions 

The  fraction  obtained  by  interchanging  numerator  and 
denominator  of  a  given  fraction  is  called  the  reciprocal  of  the 

given  fraction.    Thus,  the  reciprocal  of  7  is  - . 
.  0      a 

Reciprocal  fractions  have  the  following  property:    The 

product  of  a  fraction  and  its  reciprocal  is  unity. 

_  a    b    a  •  b    ab 

For  i:'-=l — =~i.  =  i- 

0   a    0 '  a    ab 

Fractions  are  divided  in  algebra  as  in  arithmetic.     To 
prove  this  consider  a-. — ,    where  a  may  be  either  an  integer, 


or  a  fraction. 

b    (      b\b    c 
c    \      cfc    b 

(Why?) 

(      b    b\    c 
^\^-^-c'-c)'V 

(Why?) 

-ia)S, 

(Why?) 

Therefore                    a-i—=a 
c 

c 
'b' 

State  in  words  a  rule  for  multiplying  a  whole  number  by  a 

fraction. 

EXERCISES 

Perform  the  indicated  operations:                                          \ 

.  y 

I.  x-^-. 

X 

,     loa* 
4-  ^5-^  :  ^^.  . 

J. .  ^y 

2.  ab-. — -  . 
z' 

5.  i8:vr^^:'57''. 
•^             ab 

l(>X^ 
7..    I2:X;3t . 

^           ly 

6.  (    ab):  ^\. 
sab 

7.  (iix^v^z^  —  Aax^v^z^- 

^.Rx'.'zs^:^'''y''\ 

a3b' 


1 20  First-Year  Mathematics 


8.  ('jm^pq—^^mp'q-\-2impq') 


yH 


ail 


xyz  _  x^y'^z  x'—y'  _  x-\-y 

^ab  '  a'b'  '  a+6    '     i 


X3y'z  '  xy'z  '  ^'  b+c  '  c+b 


a'b^z^    a^^b^c'  b—c^     i 

b+c~c+ 

a+b  ^    a+b 
x—y  '  x'—y' 


(1Kb'        27c'\      abc 
1 6c        lo^d/     28d3 

(m'  ^  i$mpx3\     jc36^3gr2  sia+b)  _  6x(a+b) 

8»  '    2'jn'y  /       rri'ti    '  4(x—y)  '  jaix—y) 

(x-\-i     x'--i\ 


a'— 121  ^  a  +  ij 
a*  — 4    '  a-\-2 


Sa'(x'  —  y')  _  ioa(x+yy 
^^'    6x'(a-by  ~gx(a'—b')  ' 

2Q    5(^(3x-7y)+6b('jy-3x)  ^   ^x-^y 


21. 


{Sa-6b){2X+7,y)        '  ^x'-gy""' 

iix'('jm+$n)  —  (2x'  +  i6y'){'jm  +  ^n)  _  jm  +  ^n 
i3X-4y)i2Sa'-s6b')  '  sa-66  ' 

2iox3y'  —  4$x^y3-\-ioSx'y*  .  ^5^^^* 
^^"       i2p'iq5+Sp3q'*  +  i6q3p^     ~  Ap^q^  ' 

(X        X—l\  ^  /    x      .  ^— 1\ 
X+l  X     /   '   \X+1  X     )  ' 


6. 

I  — -ya   * 

X'-l 

.8.  r* 

b  —  a 

y^—l  ' 

7- 

(I)- 

(-^)- 

X~l 

29.  

^    I— rx; 

Fractions  121 

30—26  2'(a— i)    4(1— a) 

^°'  4b-6a'  ^^'  ^.(x-iysix-i)' 

iox—8y  ii(a—x)  ^  'j(x—a) 

^^'   4y-5x  '  ^'^'    gib-y)  '^ 6{y-h) ' 

2a^*(— 3&)  cyHm—r)     dm—r) 

^  6ab  ^^     }y(x-8)     }(x-8) 


CHAPTER  XVII 

EQUATIONS  INVOLVING  FACTORABLE  FORMS 

§  58.     A  Monomial  Factor 

I.  How  long  is  each  of  the  three  parts  of  AB  ?    How  long 


j 1     is  AB  ? 

S\     30     \r  ^y  f      3d^     f        2.  Give  a  common  factor  of 
I  I     30,  25,  and  35,     Give  a  factor 

Fig.  62  of  30  +  25+35,  or  90. 

3.  Factor  these  sums  and  differences  as  the  first  one  is 
factored : 

(i)  15  +  21=3(5  +  7);  (7)  i4^  +  26:v=? 

(2)  70  +  21=?  (8)  ab+ac=? 

(3)  60-35=?  (9)  xy-xz=? 

(4)  40—12=?  (10)  xy—x=? 

(5)  i2fl  +  5a=?  (11)  x'y—x'—? 

(6)  yx+i^x^  ?  (12)  abc+aby=  ? 

4.  Factor  these  sums  and  differences  as  the  first  one  is 
factored: 

(i)  12+9  +  6=3(4+3+2);  (7)  ^ab-sac+6ad=? 

(2)  20+15  —  10=?  (8)  7c*(/— i4c':x;+42c"z=  ? 

(3)  18-6  +  10=  ?  (9)  6a'+a'd=? 

(4)  ab+ax+ad=  ?  (10)  i2cd—4cdx=  ? 

(5)  acx—acy+acz=?  (11)  i6x'y+8xy'=? 

(6)  a'b+a'x+a'=?  (12)  'jx*y—2ix'y*=? 

5.  Make  a  rule  for  factoring  sums  and  differences  when 
the  separate  terms  contain  a  monomial   (one-termed)  factor. 

6.  Solve  the  following  equations  by  factoring  the  first 
member  (side),  dividing  both  members  by  a  common  factor, 
and  then  combining  terms: 


Equations  Involving  Factorable  Forms  123 

(i)  3;c— 18=9  ;  (7)  ax—a=2,a; 

(2)  2;y+i2=8;  (8)  6^—36  =  26; 

(3)  43'-4  =  i6;  (9)  c2+ac=3ac  ; 

(4)  7x+i4  =  2i  ;  (10)  aby—abc=4abc  ; 
(5)13^+26=39;  (11)  i3«>'+26«=39n; 
(6)  9:x;— 18=9  ;  (12)  ^4X'y—i'jxy=^4xy. 

7.  Given  ^bx—acx=2ob—4ac;  to  find  x. 

'  Show  by  multiplying  that  ^bx—acx=(^b—ac)x,  and  that 
206  — 4ac =4(56— ac). 

We  may  then  write  (^b—ac)x=4{$b—ac).    Why? 
Dividing  both  sides  by  ^b—ac,  x=4. 

8.  Solve  the  following  as  7  is  solved: 

(i)  :^my  +  sdy=gm  +  isd; 

(2)  5r^-7:x:= 25^-35  ; 

(3)  ax+bx=s(ib-\-sb''  ; 

(4)  cz-sbdz^5c-i5bd; 

(5)  cx+dx—fx='jac  +  'jad—'jaf; 

(6)  by—2dy-\-5ry=6bs—i6bds  +  i3brs  ; 

(7)  wr2— i2az+&^cz=8mr/— 96a/+8&2c/ ; 

(8)  a3x+sb''x—cSdx='ja3s+:iSb''s-\-'jc^ds. 

§  59.     A  Binomial  Factor 

9.  (a+b)(x+y)=?  What  are  the  factors  of  ax-i-ay 
+bx-\-by?  How  many  terms  of  the  product  contain  a  as  a 
factor  ?    What  numbers  does  a  multiply  ? 

How  many  terms  of  the  product  contain  6  as  a  factor? 
What  numbers  does  b  multiply  ? 

Remark:  This  product  may  also  be  shown  by  a  divided 
rectangle. 

10.  (a—b){x—y)—?  What  are  the  factors  of  ax+ay 
—by—bx? 

11.  (a+b)(x—y)=?  What  are  the  factors  of  ax— ay 
+bx—by? 


124  First-Year  Mathematics 

12.  {a—b){x—y)=}  What  are  the  factors  of  ax— ay 
—hx-\-hy} 

13.  Examine  the  four  products  of  9,  10,  11,  12  and  make 
a  rule  for  finding  the  binomial  factors  of  expressions  having 
4  terms,  the  first  two  and  last  two  of  which  terms  have  a  com- 
mon factor. 

14.  Find  what  x-\-y  equals  in  the  following: 

(i)  ax-\-ay-^bx+hy=a-{-h  ; 

(2)  ax+ay+hx+hy=7^a+7^h  ; 

(3)  ax+ay—hx—by—2a  —  2h  ; 

(4)  ax+ay—hx—hy=']a—'jh  ; 

(5)  ax+ay-\-hx-\-hy'=c{a-\-h)  ; 

(6)  ax-\-ay—hx—hy=h{a—b). 

15.  To  what  is  a +6  equal  in  these  equations  ? 

(i)  ax-\ray-\-hx+hy=2,{x-\-y); 

(2)  ax—ay-\-hx—hy=c{x—y); 

(3)  ax—ay+hx—hy=i2{x—y)', 

(4)  ac+ad-\-bc-\-hd=x{c-\-d). 

16.  To  what  is  a-f  :3c  equal  in  these  equations  ? 

(i)  ab+hx-{-ay-\-xy=c(h-\-y); 
(2)  ac—2ad-\-cx—2dx=cin—2dm. 

17.  Find  X  in  the  following  : 

(i)  cx-\-dx-\-T,c-\-2t^=2{c+d)\ 

(2)  Jx+wjc— 5&  — 5w=36+3W  ; 

(3)  5r:x;— 55:x;— 2r-l-25=8r— 85  ; 

(4)  cdx—b'x—acd-\rab'  =  2acd—2db'. 

Squares  and  products  of  sums  and  differences. 

18.  (a+6)(a+6)  =  ?    What  are  the  factors  of  a'-\-2ab+b*  ? 

19.  (a— 6)(a— 6)==  ?    What  are  the  factors  of  a*  — 206+6*  ? 

20.  (a+b)(a—b)=?    What  are  the  factors  of  a'—b'? 


Equations  Involving  Factorable  Forms  125 

21.  Find  the  following  products : 

(i)  {xi-y)(x+y);  (7)  (a+x)(a-x); 

(2)  (x+y)(x-y);  (8)  (a+x)(a-^x); 

(3)  (x-y)(x-y);  (9)  (a-rx:)(a-^); 

(4)  (b+cXb-c);  (10)  (:v+3)(x-3); 

(5)  (6-0(^-c);  (II)  (^+7)(^-7); 

(6)  (6+c)(6+c);  (12)  (^-6)(^-6). 

22.  Find,  by  multiplying,  the  values  of 

(i)  (a-{-xy;    (3)  (c-yy;    (5)  (x+sY;     (7)  {y+2cy; 
(2)  (a-6)-     (4)  (r+5)-    (6)  (x-7r;     (8)  (/'^-2?)». 

23.  Find,  by  multiplying,  the  values  of 

(i)  (a+b){a—b);    what  are  the  factors  of  a'—b'? 
(2)  (a+x)(a-x);       "      "     "        "        "  a'-x'? 

(3)  ((i+y)(<^-y);     "    "   "      "     "  «^-:y^? 

(4)  (c+d)(c-d);        "      "     "        "       "  c'-d'? 

(5)  (^+:y)(^->');     "    "   "      "     "  x^-r? 

(6)  (2a+35(;)(2a— 35f) ;    what  are  the  factors  of  4a'— gy'  ? 

(7)  (x+sy)(^-sy);       "    "   "      "     "   x'-2sy'? 

(8)  (x+2cd)(x-2cd);        "      "     "        "        "     x'-4c'd'? 

24.  Show  that  the  following  are  perfect  squares  and  give 
their  square  roots: 

(i)  m'  +  2mx+x';  (6)  ^a^  +  ^ac -\- c' ; 

(2)  c^  —  2cd+d'^',  (7)  4x'  —  i2xy+gy'; 

(3)  a*  +  2flc?+(^^;  (8)  a'x''+4axy+4y'; 

(4)  x'  +  2xy+y';  (9)  jc^  — 6x+9  ; 

(5)  ae*  — 2:x;;y+>'*;  (10)  i6r'~  24rs  +  gs' . 

25.  Show,  by  multiplying,  that  the  square  of  any  binomial 
sum  or  difference  is  the  trinomial  having  two  terms  that  are 
perfect  squares  and  a  third  term  which  is  plus  or  minus  the 
double  product  of  the  square  roots  of  the  terms  that  are  per- 
fect squares. 


126  .    First-Year  Mathematics 

26.  Complete  the  following  binomials  into  perfect  trino- 
mial squares  by  adding  or  subtracting  one  number: 

(i)  a'-\-2ax;  ($)  x^-\-6x;  (9)  ^a'—zab; 

(2)  —zax+x";  (6)  x'—6x',  (10)  gy'-\-4x'; 

(3)  a'+b';  (7)  ^'+9;  (11)  9y'  +  i2xy; 

(4)  a'  —  2ac;  (8)  9— 6x;  (12)  4X'  —  i2xy. 

27.  Find  a  value  of  the  unknown,  3i?  or  %  in  each  of  these 
equations : 

(i)  x'-\-6x-{-g  =  2x+6  ;  (5)  x'  —  2ax-\-a'=x—a; 

(2)  y'—4y-\-4  =  6y—i2  ;  (6)  iv;"  — ;y'=:x;— ;y  ; 

(3)  x''  +  2xy+y'=^xy+y'',  (7)  )'='-6='  =  >'+6  ; 

(4)  :x;^  +  io5e+25=65(;+3o  ;  (8)  x^—c''=x-[-c. 

28.  (a)  Given  ^bx—acx='ja—2b,  to  solve  for  a;. 

(b)  Factoring  first  member,  x(^b—ac)='ja—2b;  for  a  fac- 
tor of  every  term  of  a  polynomial  is  a  factor  of  the  polynomial. 
The  other  factor  is  the  quotient  obtained  by  dividing  the 
polynomial  by  this  factor  of  every  term. 

7<I  —  26 
Dividing  both  members  of  (b)  by  (56— ac):    ^=^; • 

29.  (a)  Given  ax+b'=bx-{-a'.  Solve  for  x,  i.  e.,  express 
X  in  terms  of  the  other  numbers  in  the  equation. 

(b)  Transposing,  ax—bx=a'—b'; 
Factoring,  x(a —b)  =  (a+b)(a—b); 
Dividing  by  (a—b)  ,     x=a+b . 

Many  equations  containing  powers  higher  than  the  first 
power  of  the  unknown  number  are  readily  solved  by  the  use 
of  factoring. 

Note. — The  following  work  of  this  paragraph  is  to  be  de- 
veloped by  the  teacher  for  the  class. 

30.  (a)  Given  M^—4»— 12=0  ;    to  solve  for  «  : 
(b)  Factoring,  (w— 6)(«+2)=o  ; 


Equations  Involving  Factorable  Forms  127 

(c)  »— 6=0,  because  o  in  place  of  w— 6  in  (b)  satisfies 
the  equation.  (See  Chap.  XV,  p.  105,  under  prob- 
lem 8.) 

Solving  (c):  w  =  6,  one  value  of  n. 

(d)  w+2=o,  for  same  reason  as  in  (c).  w=— 2, 
another  value  of  n. 

.*.  6  and  —2  are  the  values  of  n. 

31.  (a)  Giwexi  x^+x^  —  6x=o.     To  solve  for  :x;. 

(b)  Factoring,  x{x^+x—(i)=o; 

(c)  Factoring  again  x{x -\-t,){x—2)=o; 

(d)  Therefore,  x=o,  one  value  of  x; 

(e)  Also  x-\-2,=o  and  x=  —  T),  another  value  of  x', 

(f)  Also  :x:— 2=0  and  x= 2,  a  third  value  of  rv. 

How  does  the  number  of  values  of  x  seem  to  correspond 
with  the  exponent  of  the  highest  power  of  the  unknown  in 
the  given  equation  ? 

32.  (a)  Q\vtr\.  y^-\-y^  —  6y=o.     To  solve  for  ;y. 

(b)  Factoring,  y{y^-\-y  —  6)=o; 

(c)  Factoring  further,  3'(}'+3)(:y— 2)=o, 

(d)  Then  ^=0,     one  value  of  y, 

(e)  Also  )'+3=o ,     and  )'=  —  3,  another  value  of  y, 

(f)  Also  y—2=o,     and  y=2  ,    another  value  of  y. 

How  is  the  number  of  values  of  y  related  to  the  exponent 
of  the  highest  power  of  yXn  the  equation  ? 

33.  (a)  G'w&n.  x^-\-x^=4x+^.     Solve  for  a;. 

(b)  Transposing  /i^x  and  4:  x^-{-x^—^x—^=o',  Why? 

(c)  Factoring,  (:x;+i)(a;+2)(:x;— 2)  =0  ; 

Therefore  x-'ri  =0 ,  and  x=—i  ; 
a;+2=o,  and:x;=— 2; 
rv— 2=0,     andic=+2. 

Show  by  substitution  that  —  i ,  —  2 ,  and  2  are  values  of  ;x;. 


128  First-Year  Mathematics 

§  60.     Uses  of  Factoring  in  Solving  Equations 

The  following  problems  illustrate  the  use  of  factoring  in 
the  solution  of  equations : 

1.  Verbal  Statement:  The  difiference  of  two  numbers  is  one, 
and  the  difiference  of  their  squares  is  nine,  find  the  numbers. 

Symbolic  statement: 

(a)  x-y=-i; 

(b)  x^  —  y^=(),    find  x  and  y  . 

Solution: 

(c)  Factoring  x'  —  y^  in  (b) ,  {x-'ry){x—y)=(); 

(d)  Dividing  the  equals  in  (c)  by  those  in  (a),  or  by 
substituting  i  for  {x—y)  as  per  (a),  gives  x-\ry—g. 

We  now  have  x—y=x  , 

and  x-\-y—^ .     Solve  for  x  and  y. 

2.  Verbal  statement:  The  sum  of  the  reciprocals  of  two 
numbers  and  also  the  difiference  of  the  squares  of  their  recipro- 
cals is  five.     Find  the  numbers. 

Symbolic  statement: 

(a)  ^+i  =  5  ; 
^  '  X    y    ^  ' 

Solution: 

(0  Factoring, 0+i)(i-i)=S; 

(d)  Dividing  the  equals  in  (c)  by  (a), =1  ; 

X    y 


(e)  Adding  (a)  and 

(d)i 

member  by  member : 

2_ 

X 

6,    . 

•.^=|. 

(f) 

Subtracting  (d) 

froir 

I  (a):    -=4, 

.•.:y=i. 

Equations  Involving  Factorable  Forms  129 

3.  Verbal  statement:  The  sum  of  two  numbers  is  5,  the 
sum  of  their  cubes  is  35.     Find  the  numbers. 

Symbolic  statement : 

(a)  x+y  =  s  ; 

(b)  x3  +  >;3=35. 

Solution  : 

(c)  Factoring  (b),     {x-\-y){x^  —  yx-\-y^)=2)^; 

(d)  Dividing  equals  (c)  by  equals  (a):    x^—xy-\-y''  =  ']; 

(e)  Squaring  equals  in  (a):  x^-\-2xy-\-y''  =  2^  ; 

(f)  Subtracting  (d)  from  (e):  3:x;3'  =  i8  and  xy=6', 

(g)  Subtracting  (f)  from  (d):  x^  —  2xy-\-y^  =  i  ; 
(h)  Extract  square  root  of  equals:  x—y=±_i  ; 

(i)    Add  (a)  and  (h)  :  2X  =  6  or  4  ,  ^=3  or  2  ; 

(j)    Subtract  (h)  from  (a) :  2  3/ =4  or  6  ,     ^'  =  2  or  3  . 

Therefore  the  numbers  are  3  and  2  . 

To  verify,  substitute  the  numbers  found  for  x  and  y  in  {a) 
and  (6)  and  state  whether  the  results  are  correct  equations. 

4.  Given  x^  —  t,x+2=o;  Siho  x^  —  t,x+2  =  {x—i){x—i) 
{x-\-2):    to  solve  for  x  and  verify. 

5.  Given  x^  —  2x'  —  iix-\-i2=o;  also  x^  — 2:x;^  — iix+12 
=  (x— i)(x+3)(x— 4) :   to  solve  for  x  and  verify. 

6.  Given  4:^4+9—37x^=0  ;  also  4^4— 37x^+9 =J(2x— 6) 
{2X+6){2X— i){2x-\- 1) :    to  solve  for  x  and  verify. 

7.  Given  -  +  -  =  s,     and —  =  i<. 

I  r     s     ^  r''     s'       ^ 

4        0        /2       ■^X  /2       '?\ 

Notice —=(-+-  1 1  — -  I .     Solve  for  r  and  s  and  verify. 

r^     s^     \r     s  /  \r     s  / 

The  student  will  supply  the  verbal  statement  for  numbers 
4  to  7.  In  practical  Hfe,  problems  are  first  stated  in  verbal 
form. 

Multiply  each  side  of  the  equations  by  the  L.  C.  M.  of 
the  denominators. 


130  First-Year  Mathemattcs 

8.  3 +-^ '—^ =!—^    solve  for  x. 

3926 

arv+i        -8  2:v— I         , 

0. =  — ; — ,   solve  for  x. 

2X— I     ^x^  —  1     2:x:  +  i 

10.  —7 — I — 1—=^1 1-6,  solve  for  ic. 

X-\-2       X-\-T,       X'  +  f)X 

X         <,a         2bx  ,      . 

11.  r — ^  = —,    solve  for  x. 

a-b    a+b    a'-b'' 


CHAPTER  XVIII 
FRACTIONS  INVOLVING  FACTORABLE  FORMS 

A  knowledge  of  factoring  is  often  necessary  in  operations 
with  fractions. 

§  6i.    Factoring  in  Reducing  Fractions  to  Lowest  Terms 

1.  In  arithmetic  f^  is  reduced  by  dividing  numerator  and 
denominator  by  the  common  factor  8,  obtaining  f ,  the  same 
in  value  as  |4,  but  in  lower  terms. 

2.  An   algebraic   fraction   is  reduced  in   the  same  way: 

2a' be 

—  can  be  reduced  by  dividing  both  numerator  and  denomi- 

nator   by   the   common   factor  a'b ,    obtaining  — 7 ,    having 

the  same  value  in  lower  terms. 

a* — b' 

2.  ; — r-  is  reduced  by  dividing  both  terms  by  the 

■^    a'-2ab+b'  /  &  j 

r  ,        ,      .   .       a-\-b 

factor  a—b,    obtammg  r  , 

a—b 

§62.     Factoring  in  Adding  and  Subtracting  Fractions 
In  the  addition  and  subtraction  of  fractions,  factors  are 

used  to  find  the  lowest  common  denominator. 

I.  In  arithmetic  in  finding  the  sum  of  -^  and  ^,  taking 

3X5X4,  or   60,    for   the   common   denominator,   instead   of 

20X15  or  300,  we  have 

3X3  ■+    4X7    =^^f=n^  the  sum  of  A  and  ^ 


3X5X4     3X5X4     60     60     60  20  15 

2.  In  i+T^F~TVj     taking   3X2X8,   or  48,     instead    of 
8X16X24,     or  3,072,  for  the   common    denominator   gives 

2X3X5       3X9  2X7     ^30  ,  27     14^43 

3X2X8     3X2X8     3X2X8     48"^48     48     48' 

131 


132  First-Year  Mathematics 

T      ,     ,  a      h      c       a^      b'      c'      a^'+b'+c' 

3.  Inalgebra    -H — +—=-—+-—+-—= . 

be    ac    ab    abc    abc    abc  abc 

43f  2  45c  2{x+y)  2{x—y) 


x^  —  y'     x—y     {pc—y){x-\-y)     {x—y){x-\-y')      x^  —  y^ 

Show  how — r  is  obtained  from and  also 

{x—y){x-]ry)  x'  —  y' 

how  the  succeeding  fraction  is  obtained  from  the  correspond- 
ing fraction. 

§  63.     Factoring  in  Multiplying  Fractions 

1.  Show  that  f  of  |^  of  f  is  reduced  to  y^j  by  canceUng 
the  common  factors  2,  3,  and  2  from  both  the  numerator  and 
denominator. 

2.  In  simplifying  resuhs  of  the  multipHcation  of  algebraic 
fractions  also,  we  cancel  common  factors  in  numerator  and 
denominator.     Show  that  by  canceling  common  factors 

ab^ac^db      ,  ab 

-jXriX—  reduces  to  — . 
cct    bd    ac  cd 

-?.  Show  that  in  X ,     by  canceUng   the  com- 

7x+yy       2y   '       ^  ^ 

mon  factors,  2  and  x-\-y,  the  product  is  — . 

ly 

^     a^-\-2a  —  i<     a'  +  Ta— 44 
4.  In  — — ^X ,     factormg  gives 

(a  +  s)(a-T,)      (a  +  ii)(a-4) 


(a  +  ii)(a-3)      (a  +  5){a+4)  ' 

Show  that  by  canceling  the  common  factors  a +  11,  a +  5,  and 

«_3weget^. 

5.  Show  by  multiplying  that  (a-|-5)(a— 3)=a^  +  2a  — 15  ; 
that  (a  +  ii)(o— 3)=a*  +  8a— 33  ;  that  (a-|-ii)(a— 4)=a^-f-7a 
—44;    and  that  (a  +  S)(o+4)  =0^+9^  +  20. 


CHAPTER  XIX 

FACTORING 

The  factors  of  a  given  number  are  the  numbers  which 
multiplied  together  will  produce  the  given  number.  The  pro- 
cess of  factoring  is  therefore  the  inverse  to  the  process  of 
multiplication.  In  this  chapter  we  consider  only  factors  which 
are  non-fractional  (integral)  and  which  do  not  involve  radical 
signs  (i.  e.,  which  are  rational  numbers).  It  is  clear  then  that 
factors  of  a  number  will  divide  it  without  a  remainder.  A 
prime  number  is  a  number  which  has  no  factors  except  itself 
and  unity.  A  prime  factor  is  a  prime  number.  A  number 
has  only  one  set  of  prime  factors. 

§  64.     Monomial  Factors 

I.  Problem:   Factor  ^a^b  —  i2ab'. 

Explanation. — Since  ;^ab  is  a  divisor  of  each  term,  it  is 
a  divisor  of  the  whole  expression,  and  dividing  306  into  the 
expression  (each  term  in  succession)  we  obtain  the  quotient 
a— 4b.  Then  306  and  a— 4b  are  the  factors  of  ^a'b  —  i2ab', 
or  ^a'b  —  i2ab^=;^ab(a—4b). 

Test  by  multipUcation: 

a— 46 

^a^b  —  i2ab^ 

Principle. — A  number  is  equal  to  the  product  of  all  oj  its 
prime  factors  for  all  values  of  its  letters.  Such  an  equality  is 
called  an  identity. 

By  this  principle,  we  can  test,  or  verify  the  correctness  of 
the  factors.  If  the  factors  of  T,a^b  —  i2ab^  are  T,ab  and  a— 4, 
then  7,a'^b  —  i2ab^  =  T)ab{a—4b)  for  all  values  of  a  and  b  (by 
the  principle) .  The  two  expressions  are  therefore  equal  for 
the  values,  a =2,  and  b  =  i. 

133 


134 


First-Year  Mathematics 


Test  by  substitution: 

For  a  =  5  and  b  =  i; 

3a='6-i2a6='=3X5'Xi-i2X5Xi^  =  i5  ; 
and  3a6(a-46)  =3X5X1(5 -4)  =i5- 

Therefore  the  expression  is  correctly  factored. 

Verify  by  substituting  other  values  for  a  and  b. 

2.  Factor  the  following  expressions  and  test  results  by  both 
multiplication  and  substitution: 


(I) 

8x3>;='+4x=')'3; 

(2) 

T^x^y''  —  2xy—T)Xy^\ 

(3) 

I  $a^x — I  oa^  y + ^a^z ; 

(4) 

2)2a^b^—ab^; 

(5) 

6^3^2+3;x;*;y3  +  35c:3;4; 

(6) 

Sx''y^  +  i6xyz  —  24x'y''z'; 

(7) 

2a='6='+a3j3-|-aj4; 

(8) 

Sa^b^'c''  -4a'b3c3  +a'b'c3 ; 

(9) 

i^x'-{-2oxy+$y^; 

(10) 

i6a^b'+4Sa'b-i6ab-\-8a. 

Reduce  to  lowest  terms  the  fraction: 

50=^6  —  10^6^ 

i^abc  +  2oa'b' 

4.  Solve  for  x  the  equations : 

(i)  ax+bx=ac+bc  ;  (3)  ax+bx=cx+d', 

(2)  abx—abd=cx—cd;  (4)  ax+bm=am-^bx . 

5.  Factor   30+36  +  50  +  5^- 
Solution: 

30  +  36  +  50  +  56=3(0  +  6) +  5(a  +  6) 
=  («  +  ^)(3  +  5). 
The  method  of  factoring  used  in  this  problem  is  called  the 
method  0}  grouping. 

6.  Factor  ac  +  bc+ad+bd. 


Factoring  135 


Solution: 

ac+bc+ad+bd=c(a+b)  -\-d{a+b) 
=  ia-\-b)(c+d). 
Test  by  multiplication: 

c+d 
a-\-b 


ac-\-ad+bc-\-bd. 
Test  by  substitution: 

Leta  =  i;6  =  2;     c=2,;  d=/[. 

Then  ac+6c+aJ+W  =  iX3  +  2X3  +  iX4  +  2X4 

=3  +  6+4  +  8  =  21, 
and  (c  +  (/)(a  +  &)  =  (3+4)(i+2)=7X3  =  2i. 
Therefore,  ac+bc+ad-\-bd  =  (c-\-d){a+b) . 

7.  Factor    i4x^—6x^  —  2ix-j-g. 
Solution: 

14^3— 6rc^  — 2  ix+9  =  2x^(7:x;— 3)— 3(7:^—3) 
=  {2x^-3X7^-3) ' 
Test  by  both  multiplication  and  by  substitution. 

8.  Resolve  into  factors  the  following  expressions  and  test 
results  by  both  methods: 

(i)  a^—ax+ab—bd;  (6)  ;^ax+7,ab-\-2x^-\-2bx-\-b+x  ; 

(2)  oc^  +  ^x^+x3  +  ^x  ',  (7)  4x3+4x—4X^z—4z; 

(3)  6x^—gx—ioxy+i^y;    (8)  i+r—r^xy—r^xy; 

(4)  2x3+x''  +  6x+^  ;  (9)  x'—x^-\-i—x; 

(5)  3^<^+3^^~5c~5^  ;       (10)  (a  +  w)(c+w)— 2«(a+w) . 

9.  Reduce  to  lowest  terms  the  following: 

X4  — 2X3+7^+14 

2^3— 4x*  +  65£:  — 12  * 

10.  Find  the  product  of  the  fractions 

x'—xy—6x+6y  ^ac  —  ^bc 

ax—bx—ay+by  xy—6y 


136  First-Year  Mathematics 

II.  Solve  for  x  : 

(i)  ax+bx=ad-^bd+ac-\-bc  ; 
(3)  ^a'bx-}-i2bcd=/^ab'x+gacd  ; 
(3)  ax—bc—ac=ab-\-bx. 

§  65.     The  Perfect  Square 

1.  Factor  a* +  2a&+6». 
Method  0}  grouping: 

a'  +  2ab-\-b'=a''-^ab+ab-\-b' 
=a(a-\-b)+b(a-\-b) 
=  (a-{-b)(a+b). 

Test  by  multiplication  and  by  substitution  that  (a+b)(a  +  b) 
=a'-\-2ab+b'. 

2.  Factor  49x3  — 154^3^-^-1 213/2. 
Method  of  grouping: 

^()X^  —  ii^^x^y-^i2iy^=^go(fi  —  ']']x^y—']']X^y-\-i2iy^ 

=  ']X^{']x^  —  iiy)  —  ii  y(jx^  —  1 1  j) 
=  {'jx^  —  iiy){']x^  —  i\y) 

Test  by  multiplication  and  by  substitution. 

Method  of  inspection:  We  have  learned  (see  pp.  107,  125, 
and  126)  that  when  a  binomial  is  multiplied  by  itself  the  result 
is  a  trinomial  which  consists  of  the  square  of  the  first  term  of 
the  binomial  plus  or  minus  (according  as  the  binomial  is  a 
sum,  or  a  difference)  twice  the  product  of  the  first  and  second 
term  of  the  binomial  plus  the  square  of  the  second  term  of 
the  binomial. 

Then  any  trinomial  in  which  two  terms  are  perfect  squares 
(and  positive)  and  the  other  term  is  plus  or  minus  twice  the 
product  of  the  square  roots  of  those  two  terms,  is  the  square  of 
the  sum  or  difference  of  those  two  square  roots  according  as 
the  third  term  is  plus  or  minus. 


Factoring  137 

3.  Add  a  third  term  to  make  perfect  squares  out  of  the 
following  and  factor: 

(i)  m^-\-n^;  (5)  m^-^-tmn;  (9)  i6^^  +  s6^;y^; 

(2)  w^  +  2w»;  (6)  n^+8wn;  (10)  r^+ys; 

(3)  2W»+»2.  ^y^  9a^62+4C^;  (11)  x^  +  S:^; 

(4)  m'+4n^;  (8)  9:x;2+30^3:x; ;  (12)  x^-\-bx. 

4.  Factor  the  following  expressions,  and  test  each  result 
both  by  multiphcation  and  substitution. 

(i)  ^^  +  6^/+9/^;  (s)  c'-i6c-\-64; 

(2)  4m'  —  i2am+ga^;  (6)  x'°+30xs  +  225  ; 

(3)  25+8or+64r^;  (7)  49  — i4o»^  +  ioo«4; 

(4)  i2ia^  +  i98a>'+8i>'^;  (8)  a^b^c'' -\-Sabc  + 16  ; 

(9)  (u-\-vy+4t(u+v)-\-4P; 
(10)  w^  — 2w«+w^  +  6aw— 6aw+9a* . 

5.  Factor  the  following  by  any  method  and  test: 

(i)  a''k3-2abk3  +  b''k3; 

(2)  x^y+x^y^—x^y'—xy*; 

(3)  ax^+ax'+ax+a  ; 

(4)  (x+yy  '  (x-y)-(x-yy  ■  (x+y); 

(5)  a''+¥-\-c^  +  2ab-{-2ac  +  2bc; 

(6)  x^— 6x>'+93'^  — 2:!cz+6jz+z2; 

(7)  ga^m+ga'n+i2am  +  i2an+4m+4n  ; 

(8)  x^—6xy+gy'  —  2xz  +  6yz+z^. 

6.  Reduce  the  following  fractions  to  lowest  terms : 

,  ,         ^a'm  +  2b'm  ,  ^  ar^s'+br's^'+cr's' 

(0  „..  ,  „..;..  ,  ./..  5  (3) 


9a44-i2a^6^+464  '  ^"^^  a^+6^+c^  +  2a6  +  2ac  +  26c  ' 

,  .      2ic^  +  iocd+d'  ,  ,       2ax—(iX—2ay-\-(iy 

(2)   ,^,  ,  ^^  ,  ■.^-  ,  .^ ;  (4)  -'-'-. 


2xS  +  2r)c;4+3:x;34-3:x:*+4x+4 ' 
7.  Solve  the  following  for  the  required  numbers: 


138  First-Year  Mathematics 

(i)  ax+;^bx=a^  +  6ab-\-gb^;    x  required. 

(2)  2mr—iomr=i^mr+4ns  +  6ns—^ny;     y  required. 

(3)  3ut-\-6vh3-{-^wt=<)uh3  +  2vt-\-i^'wh^;    /required. 

(4)  •ja'b''q+42a'b'c''d'=4ga'^b'^-\-^c''d'q+gc'^d'^  ;    g^  required. 

(5)  iia'w—i;^b'w—22a'm-\-26b^Tn=o;     w  required;      w  re- 
quired. 

§  66.     The  Difference  of  Two  Squares 

I.  Factor  a'—b'. 
Method  of  grouping : 

a^-b'=a^-ab+ab-b'  (Why?) 

=a(a-b)+b{a-b)  (Why?) 

=  (a  +  b){a-b).  (Why?) 

Test  in  two  ways. 

Method  by  inspection:  We  know  from  the  chapter  on  multi- 
pUcation  that  when  the  sum  of  two  numbers  is  multipUed  by 
their  difference,  the  result  is  the  difference  of  their  squares,  viz. : 
2k  -I-3W 
2k  — 3W 

4k^-\-6km 

—  6km—gm^ 


4k'  —gm' 

Therefore,  the  difference  of  two  squares  factors  into  the  sum 
and  difference  of  the  square  roots  of  the  two  squares. 

2.  Factor  the  following  expressions  and  test  results  by 
both   methods : 

(i)  x'  —  y^;  (8)  4aHd—25c5d; 

(2)  i6k'-2sl';  (9)  {r+3sy-i6t^; 

(3)  r^-s'';  (10)  b'-{3a  +  2cy; 

(4)  Siw*  — i6»'*f4j  (11)  i6a^  — (2w— 3»)*; 
{<,)  P'q^'r-r;               ~  (12)  a''-2ab  +  b''-c'; 

(6)  ax'^  —  iooa;  (13)  i—a^  —  2ab—b'', 

(7)  16-253'^;  (14)  (^a-2by-(2c-sd)'; 


Factoring  139 

(15)  a^  +  2a  +  26c— &^— c^  +  i  ; 

(16)  ga^  —  i2ab-\-^h^  —  i6x^—^xy—y^; 

(17)  x'^+x^y^  +  y'^; 

(Suggestion :  x'^ + x^y^  -\-y'^=x'^  +  2x'y^  +  J''*  —  x^y^) . 

(18)  a'^-'ja^'b^'  +  b'^; 

(Suggestion:  a4  —  'ja'b^  +  b4=a'*  +  2a'b'+b'^—g^ab'). 

(19)  i6x'^  — I 'J x^y^  +  y^;         (21)  x'^+x'  +  i  ; 

(20)  25X''4-3i^^3'^  +  i6>''*;      (22)  a*3£;^+o^:xr**+a* . 

3.  Reduce  the  following  fractions  to  lowest  terms: 
mx'—my^ 


(0 

(2) 

(3) 
(4) 
(5) 


ax—ay+bx—by  ' 

a'+4ab+4b'—gc''  ^ 

a'+4b^+gc'+4ab  —  6ac  —  i2bc  ' 

i—a''  —  2ab—¥ 
i+a2  +  6^  +  2a  +  26  +  2c&  ' 


(^^-a=')(.'V='-6^)* 
4.  In  the  following  problems  perform  the  indicated  opera- 


tions : 


^^^    a^-8a  +  i6      a^-64  ' 

4X^  —  i2xy+gy^     x'  +  a^+b^  +  2ax+2bx+2ab 
x^—a^  —  2ab—¥  4ax'—gay' 

qs—ab'i     ^       gi+a^'b  +  ab^+b^      ^ 
120^  —  7564  *  24a^6  — 120C&3  +  1506S  ' 

2abx+2aby+2a'b'x+2a^b^y  ^  4x'  +  i2xy+gy' 
x^  +  2abx^+a^b'x^  '        mx^+nx' 


(3) 
(4) 


04—^4  wr+w^      .  mc'+mb' 

^^^  a^¥+2abFTb^     a^^^^2aF+b'^'      ¥ 


140  First-Year  Mathematics 

5.  Solve  the  following  equations  for  the  required  number: 
(i)  ak  +  bk==a'—b' .     Solve  for  ^. 

(2)  a^m—b'm=an-\-bn.     Solve  for  w;   solve  for  «. 

(3)  k^y+2k^y—z=k^z  —  y.     Solve  for  ;y;   solve  for  2. 

(4)  m'*—6m'n^+n*=m^r+2mrir—n^r.     Solve  for  r. 

(5)  a'v-\-acv-\-^abv=a^t—c't.     Solve  for  7; ;  solve  for /. 

§  67.     The  Trinomial  of  the  Form  x^+ax+b 

1.  Factor  :x;"  +  7x+i2. 

Solution  by  separating  and  grouping: 

x'  +  'jx-}-i2=x^-\-;^x+4X+i2  (Why?) 

=x(:v+3) +4(^+3)         (Why?) 
=  (^+3)(^+4).  (Why?) 

Test  in  two  ways. 

2.  Factor  5(;^—x— 30. 
Solution  by  grouping. 

x^—x—^o=x^—6x+^x—^o  (Why?) 

=x(x-6)  +  s{x-6)  (Why?) 

=  {x-6)(x+s)-  (Why?) 

Test  in  twoiways. 

3.  Notice,  in  problem  i,  viz.: 

(x'  +  'jx+i2)  =  {x-\-^){x+4) 
that  the  sum  of  the  numbers  +3  and  +4  connected  with  x 
in  the  two  factors  is  +7,  which  is  the  coefficient  of  x  in  the 
trinomial,  and  their  product  is  +12,  the  term  of  the  trinomial 
not  containing  x  (i.  e.,  the  absolute  term).  Is  this  true  in 
problem  2  ? 

This  is  proved  to  be  true  in  general  as  follows: 

X  +a 

X  +b 

x^+ax 

+  bx+ab 

x'+ax+  bx  +ab=x'  +  (a+b)x+ab. 

Therefore  x^+  (a+b)x+ab  =  {x+a)(x-\-b). 


Factoring  141 

The  sum  of  +a  and  -\-h=a-\-b,  the  coefficient  of  x  in  the  tri- 
nomial. The  product  of  +fi^  and  -\-h  =  -\-ab,  the  term  of  the 
trinomial  not  containing  :x:  (i.e.,  the  absolute  term).  This 
gives  us  a  method  of  factoring  by  inspection. 

4.  Factor  x^-\-i2X+;^^  by  inspection. 

Solution. — Find  two  factors  of  +35  whose  sum  is  +12. 
They  are  +5  and  +7.  Then  the  factors  of  x^-{-i2X-\-;^<,  are 
(:x;-|-5)  and  (x+7). 

5.  Factor  x'  —  t,x—^o  by  inspection. 

The  factors  of  —40  whose  sum  is  —3  are  —8  and  +5. 
Therefore  {x^  —  t,x—^o)  =  {x—?>){x-\-^). 

6.  Factor  the  following  expressions  by  inspection : 

(i)  x^  +  T,x-\-2  ;  (10)  /^  — 14/  — 51  ; 

(2)  w^  +  5w4-6  ;  (11)  r^  — 2^5—3235^; 

(3)  ^^  +  i2^L-|-35L^;  (12)  v^—6vw—giw^  ;    . 

(4)  y^  +  i4y+45  ;  (13)  a^b'-4abc'-i6sc'^; 

(5)  h'-iT,h+4o;  (14)  z4-ioz^-24; 

(6)  a^+8a-2o;  (1$)  y'-i2y-8s  ; 
(7)r+3^-i8o;  {16)  d^-jd- so; 
(8)  b'-{-igbc+4Sc';  (17)  p'-p-go; 

(g)  q^+q'-42;  (iS)  (a  +  by-y(a  +  b) +  10] 

(19)  a^  +  2a6-f-6^+3a+36-io; 

(20)  a^—6ab+gb^  +  'jac  —  2ibc—44C^. 

7.  Factor  the  following  by  any  method  and  test  results  in 
one  way : 

(i)  a^bc3-2abci-Sbc3; 

(2)  b'm-\-b^n  —  i2btn  —  i2bn—4^m—4$n; 

(3)  p*-26p^  +  2s; 

(4)  p*-i6; 

(5)  2ax+2ay  —  2az  +  2bx+2by—2bz; 

(6)  v^^—w^^  into  five  factors; 

(7)  m'—2mn-^n^—c'—d'—2cd; 

(8)  8^3_6;c2-28:x;-|-2i; 


142  First-Year  Mathemakcs 

(9)  3k'-6kl+^l'  +  6km-6lm; 
(10)  x'—4y'-{-i2yz—gz'. 

8.  Reduce  the  following  fractions  to  lowest  terms: 

x'-\-ax-\-bx+ab  2x+bx—3b—6 


,  .   2ajg+3&x+ 4^  +  66.  .      w^  — i2m+32 

^^^    x='+:x;(ft  +  2)  +  26   '  ^"^^    m='  +  2W-24  ' 

§  68.     The  Trinomial  of  the  Form  ax'+bx-^c. 

1.  Factor  6x^  +  17x4-12. 

Solution  by  separating  and  grouping: 

6x''  +  i'jx-{-i2=6x'-{-Sx+gx+i2  (Why?) 

=  2.r(3X+ 4) +3(3^+4)       (Why?) 

=  (3X+4)(2^+3)..  (Why?) 

Test  in  two  ways. 

2.  Factor  8z*— 37Z  — 15. 

Solution  by  separating  and  grouping: 

82^—372  —  15  =  82^—402+32  —  15  (Why?) 

=  82(2-5) +3(2-5)  (Why?) 

=  (2-5)(82+3).  (Why?) 

Test  in  two  ways. 

3.  Factor  6a^  — 7a— 3. 

Solution  by  substitution:    Multiply  the   expression  by  6, 
and  write  the  result  in  the  following  form: 

(6a)^-7X6a-i8. 
Put  2= 6a,  and  we  have 

2^-72-18, 
which  has  two  factors,  (2—9)  and  (2  +  2).     Then,  since  we 
have  multipUed  the  expression  by  6, 

,  ,  (2— o)(2  +  2)     (6a— 9)(6a  +  2) 

6a--7a-3  =  ^ ^ ^=^ ^^^- ^=(2a-3)(3a  +  i) 

4.  Factor  S>'^  +  i3>'— 6. 


Factoring  I4j 

Solution  by  substitution: 


5 
z='  +  i3Z— 30 

5 
(z  +  i5)(z-2) 


(where  z  =  ^y)\ 


5 
__(5y+i5)(5>'-2)  . 
5 

=(:v+3)(5>'-2)- 

5.  Factor  the  following  expressions,  and  test  each  solution: 

(i)  2x^  +  11^+12  ;  (7)  7^^  +  123^  —  54  ; 

(2)  Sc'+46c-i2  ;  (8)  12^^  +  315^-155^  ; 

(3)  3^^  — 17X+10;  (9)  ^m'  —  2gmn+;^6n'; 

(4)  8z^-37z-i5  ;  (10)  10^^-23^-5; 

(5)  5^'-38^+2i  ;  (11)  66^-296+35  ; 

(6)  I la^  — 23^6  +  26^;  (12)  6/^—/— 77  ; 

(13)  6(a+br-7(a+b)-s; 

(14)  4X^+8x>'+4)'^  +  i3:x;+i3)/+3  ; 

(15)  3c^— 6c(/+3(/^  — 2c  +  2(i  — 5  . 

6.  Factor  the  following  by  any  method  and  test  the  results : 
(i)  2c'xy—i2cdxy+6d^xy;  (6)  a'—b'+a—b; 

(2)  3x3  +  2x^— 9^- 6  ;  (7)  i6x4  — 81  ; 

(3)  X'*— 2x^  +  1;  (8)  o9  — 256a; 

(4)  6a3— 3oa*6+36a6';  (9)  4a4—ga'  +  6a  —  i; 

(5)  a^-6»— c»  — 26c+a+6  +  c;  (10)  a^^b^-^ab^-^jb^. 

7.  Reduce  the  following  fractions  to  lowest  terms: 

20^  +  170  +  21  am'— am  — 20a  _ 

^  ^  30^  +  260+35  '  2W^  — 7w  — 15    ' 

6c?^-5(^-6  isk'  +  kl-2l'  ^ 


144  First- Year  Mathematics 


\3)  „„  .,2     rmTjTTTT'  '  ^  ) 


2ay^  —  i^ay-{-2ia  9^^— 36X— 126^ 

8.  In  the  following  problems  perform  the  indicated  opera- 
tions: 

(I) 


(2) 
(3) 
(4) 


iSc^'  +  c  —  s     2c='  +  7c— 4  ' 

iobx-\-T,b^  +  ^x'  _  (T^b+x)x 
iobx—^b^  —  7,x'  '  (x  —  T,b)b  ' 

y^  +  y  —  l       y^  —  y  —  i 

2y'-y-3    zy^-y-A  ' 

y'^  —  \/^y-\-2\  ^  (>'2+4>'— i2)X(>'w  +  5y) 


w^+9w— 36     (w^  +  2w— i5)X(>'w  +  6w) 

9.  Solve  the  following  first  degree  equations  for  the  de- 
sired number: 

(i)  2aw^  — 5am  +  3w=6»^  +  6»  — 26.     Solve  for  a;  fori. 

(2)  \<)kx'^-\-kxy—(iky^=^x^-\-lxy  —  2y'^.     Solve  for  ^. 

(3)  tx'^—tx  =  2ot-\-2x'^  —  ']x—\<).     Solve  for /. 

(4)  2a'^x-\-\'jax-\-2\x—'^a'^y  —  2tay—2,'^y=o.  Solve  for  x ;  for  ^r. 

(5)  mx'^-\-i^m-\-\ynx=Zmx-\-2nx^-\-2\n.    Solve  form;  for  «. 

10.  Solve  the  following  quadratic  equations  for  the  required 
number : 

(i)  2x^  —  13^+6=0.     Solve  for  X. 
Solution  : 

Factoring,  (2rx;  — i)(:x;  — 6)=o; 

Then,  by  the  principle  stated  on  p.  105  under  problem  8; 

either  2^;— i  =0,  whence  x  =  \, 

ox  x—t=o,  whence  jc  =  6. 

and  the  two  solutions  of  the  quadratic  equation 

2x^  — i3X+6==o,  are  x  =  t  and  x=\. 

(2)  6:v^  +  i7:x;+i2=o.     Solve  for  :x:. 

(3)  6x^  =  w:x;+35w^ .     Solve  for  w  ;   for  :». 


Factoring  145 

(4)  7,6k''  =  i^kl-\-6l'' .     Solve  for  k  ;  for  /. 

(5)  5>'^  +  i3>'=6.     Solve  for  y. 

(6)  24m^  +  i$n'=^Smn.     Solve  for  m;   for  n. 

(7)  2c^+6d'  =  i;^cd.     Solve  for  c  ;   for  d. 

(8)  8a'  +  i4ab  =  i^b^ .     Solve  for  a  ;  -for  b. 

(9)  i4:x;*  +  2i)'*  =  55yx.     Solve  for  :x; ;  for  y. 

(10)  Sk^l^  =  22klm  +  2im^ .     Solve  for  ^^,  for  ^,  for  I  and  w. 

§  69.     The  Sum  or  Difference  of  Two  Cubes 

1.  By  multiplying  prove  that 

x3-\-y3  =  {x-\-y)(x'—xy+y')  , 
and  th&tx3  —  y3  =  (x—y)(x'-\-xy+y^)  . 

Therefore,  the  sum  of  two  cubes  has  for  one  factor  the 
sum  of  the  numbers  that  are  cubed,  and  the  other  factor  is 
the  square  of  the  first  number  of  the  factor  minus  the  product 
of  the  first  and  second  numbers  plus  the  square  of  the  second 
number. 

Formulate  a  similar  statement  for  the  difference  of  cubes. 

2.  Factor  the  following  expressions: 

(i)  64a3  +  2'jb3. 
Solution. 

64a3 4-2763  =  (4a -|-3&)(i6a'  —  i2a6+96*) . 

The  expression  is  the  sum  of  two  cubes  (4a)  3  and  (36)3, 
therefore  one  factor  is  the  sum  of  the  numbers  cubed  (4a  and 
3&),  and  the  other  factor  is  the  square  of  the  first  number, 
(4a)'  =  i6a',  of  the  first  factor,  minus  the  product  of  the  first 
and  second  numbers,  —  (4^X36)  =  — i2a&,  plus  the  square  of 
the  second  number  (36)^=96'. 

(2)  8x3  — 125 J3. 

Solution: 

8x^'-i2^y3  =  (2x—$y)(4x'  +  ioxy+2$y')  . 


146  First- Year  Mathematics 

(3)  W3  +  27W3;  (q)    8v^^-\-2'JV^^  ; 

(4)  Sc3-d3;  (10)  'j2ga^  +  2i6c^; 

(5)  343-^^;  (11)    5l2C3-27(i3; 

(6)  /3+64;  (12)  ^3L3+343  ; 

(7)  8:^^8  +  272;";  (13)  («;+3)3-a3; 

(8)  2'ju9w^  +  i;  (14)  (5W— w)3+c3. 
3.  Factor  the  following  and  test: 

(i)  a^-b^;  (6)  a^-b^; 

(2)  a^-\-b^;  (7)  a3+b3-\-a+b; 

(3)  a"-6";  (8)  a3-b3-\-a-b; 

(4)  a" +6";  (9)   af3-3;3_33i;3,(5(;-3;)  ; 

(5)  a9+b9;  (lo)  a4_a3j+aft3_64. 

'  4.  In  the  following  problems  perform  the  indicated  opera- 
tions and  reduce  to  lowest  terms: 
3(;3  — 27 


(I) 


iC'  +  2X  — 15 


(2)  zr^:z+: 


x—2a    x^—Sa^    x'  +  2ax+4a' 
x'—xy+y'    x^  —  y^     {y—xY 


rx;2+3t:;y  +  3'"     a;3-f  3^3     (^+3;)2 

Jf3  — 8;y3         ^C^— JCJ  +  y*         ;X;3— 5C>'* 

x'—xy     x'-\-2xy+4y'      x^  +  y^  ' 

§  70.    The  Factor  Theorem 
Divide  ax'-}-bx-\-c  hy  x—r. 

x—r)ax'+bx+c(ax-{-b+ar 
ax'—axr 


bx+arx 
bx—br 

arx+br+c 
arx—ar' 
Remainder,  ar'  +  br-\-c. 


Factoring  147 

2.  What  is  the  remainder  obtained  by  dividing  kx^+lx' 
+mx+n  by  x—t? 

3.  What  is  the  remainder  obtained  by  dividing  x'+px 
+q  by  x+k  ? 

4.  What  is  the  remainder  obtained  by  dividing  kx^+lx" 
+mx+n  by  x+a  ? 

5.  We  find  that  in  these  four  problems  the  remainder  in 
each  case  is  the  expression  obtained  by  replacing  x  in  the " 
dividend  by  the  negative  of  the  number  connected  with  x  in 
the  divisor.     Verify  this  in  each  problem  by  substituting. 

In  problem  i,  if  ar'-\-br+c  had  turned  out  to  be  zero, 
there  would  be  no  remainder  and  x—r  would  be  an  exact 
divisor  or  a  factor  of  ax^+bx+c. 

6.  If  x3  —  ^x'  +  2X+4  be  divided  by  rx;— 3  what  remainder 
would  be  expected  from  the  above  results  ? 

Verify  your  answer  by  actual  division. 

7.  If  ^x'  —  ^x'+4x-^i  is  divided  by  x+2,  by  substitution 
what  is  the  remainder?    Verify  by  division. 

8.  Show  that  xi—^x^-i-2X—6  has  a  factor  x—t,  by  using 
the  above  principle  (without  dividing). 

9.  The  proof  will  now  be  given  of  the  theorem  we  have 
been  using,  i.  e.,  that,  if  in  a  given  expression  containing  x 
(call  it  for  brevity  an  expression  in  x),  r  is  put  for  x,  the 
expression  in  x  reduces  to  the  remainder  J^,  obtained  by 
dividing  the  expression  in  :»  by  x—r. 

Principle. — In  division  there  is  always  a  divisor  d,  a 
dividend  D,  a  quotient  Q,  and  a  remainder  R  (sometimes 
zero),  and  the  [allowing  relation  connects  them. 

D=QXd+R. 

Then,  if  the  expression  in  x  is  divided  by  x—r,  by  this 
principle  we  have  the  expression  in  x  =Q{x—r)-\-R. 

Substitute  on  both  sides,  for  x,  the  number  r,  then  expression 
inr  =Q{r-r)+R=QXo^-R=R . 


148  First-Year  Mathematics 

Therefore,  R,  the  remainder  found  by  dividing  the  expres- 
sion in  ;x;  by  :v— r,  equals  expression  in  r,  i.  e.,  the  given  ex- 
pression withf  put  in  place  of  x. 

10.  Factor: 

(1)  x^  —  <,x'  —  2X+2^;  (5)  x'^~7,x^  —  2ix^+^2)^-\-6o; 

(2)  ic3-f  8  ;  (6)  x*—x$x'-\-iox-\-2i\', 

(3)  ^3— 7^^  +  7^+15  ;  (7)  x^-\-2ax'-V<,a'x-'r^a^  ; 

(4)  x^-i  ;  (8)  >r3+64. 

11.  Solve  the  following  equations  : 

(i)    X^+^X''  —  \T,X—1<)=0. 

Solution:    x=—i,  reduces  the  expression  on  the  left  to 
zero,  then  x-{  —  i),ox  x-\-i  is  a  factor . 
By  dividing  we  get: 

x^  -\-  2,x^  —  i^x—i<^  —  (x-\-i){x^  -\-2X—l$) 
=  {x+i){x-7,){x+s)  ; 
Then  (x+i)(x-3)(:v+5)=o, 
Whence  :»=— 1,3,— 5.     Why? 

(2)  yfe34.2^2+4)fe-}-8=o.     Solve  for  ^. 

(3)  w3— 6w*  +  iiw— 6=0.     Solve  form. 

(4)  4)'4+323'3+83>'"-t-76)'-H2i=o.     Solve  for  y. 

(5)  a*—iia^-\-/^j^a^  —  ']6a-\-^%=o.    Solve  for  a. 


CHAPTER  XX 
RATIO,  PROPORTION  AND  SIMILARITY 
§71.     Examples  and  Definition 
The  ratio  of  6  to  3  is  f ,  or  2  ;  of  3  to  4  is  f  ;   of  a  to  6  is  -  . 

The  ratio  of  6  to  3  is  sometimes  written  6:3;  of  3  to  4,  3 : 4  ; 
and  of  a  to  &,  a  :  & . 

Definition. — The  ratio  of  any  number  to  another  number 
is  the  quotient  found  by  dividing  the  first  number  by  the  second. 
Thus  f  is  the  ratio  of  2  to  3.  Any  fraction  may  be  regarded 
as  an  expression  of  the  ratio  of  its  numerator  to  its  denomina- 
tor. 

In  all  problems  and  exercises,  answer  all  you  can  orally. 
Use  pencil  and  paper  only  when  necessary. 

§  72.     Exercises  and  Problems 

1.  Write  in  two  ways  the  following  ratios: 

(i)  5  to  20  ;  (7)  10  to  50  ;  (13)  a+b  to  c  ; 

(2)  20  to    5  ;       '       {2>)  xio    y\  (14)  x+y\.ox—y\ 

(3)  15  to  20  ;  (9)  ciod;  (15)  a—bioc+d', 

(4)  18  to  25  ;  (10)  dto  c,  (i6)  ax-\-ay  to  a  ; 

(5)  25  to  18  ;  (11)  mtor;  (17)  $x+2  to  ab  ; 

(6)  50  to  10  ;  (12)  y  to  X  ',  (18)  a+b'to  x  . 

2.  A  rectangle  is  6'  by  18'  and  another  7'  by  18'.  What 
is  the  ratio  of  their  areas  ?  Of  their  lengths  ?  Of  their 
widths  ? 

3.  One  triangle  has  a  base  of  24'  and  an  altitude  of  10'; 
another  has  a  base  of  12'  and  an  altitude  of  10'. ,  What  is 
the  ratio  of  their  areas  ?     Of  their  bases  ?     Of  their  altitudes  ? 

4.  What  is  the  ratio  of  the  area  of  a  rectangle  1 2^X20'  to 
that  of  a  triangle  12'  high  by  20'  long  ?  What  is  the  ratio  of 
their  lengths  ?     Of  their  heights  ? 

149 


150  First- Year  Mathematics 

5.  How  do  two  numbers  whose  ratio  is  i  compare  in  size  ? 
•  6.  What  is  the  ratio  of  areas  of  a  triangle  and  a  rectangle 
whose  bases  are  b  and  altitudes  a  ? 

7.  What  is  the  ratio  of  the  cost  of  5  yd.  of  silk  at  $1 .  50 
to  50  yd.  of  cotton  at  12^  cents  ? 

8.  What  is  the  ratio  of  one  yard  to  one  foot?  Of  i  yd, 
to  I  in.  ?  Of  I  yd.  to  6  in.  ?  Of  3  yd.  to  3  in.  ?  Of  3  yd. 
to  3  feet  ? 

9.  What  is  the  ratio  of  i  lb.  to  i  oz.  ?  Of  i  oz.  to  5  lb.  ? 
Of  I  ton  to  500  lb.  ?     Of  5  lb.  to  5  ounces  ? 

10.  What  is  the  ratio  of  i  mi.  to  i  yd.  ?  Of  i  mi.  to  i  ft.  ? 
Of  I  mi.  to  880  ft.  ?    of  I  mi.  to  880  inches  ? 

Problems  8,  9,  10  illustrate  that  magnitudes  must  be  ex- 
pressed in  the  same  unit  before  their  ratio  can  be  expressed 
as  a  single  number. 

11.  How  does  the  ratio  of  the  areas  of  the  rectangles  of 
problem  2  compare  with  the  ratio  of  the  widths  ?  How  do 
the  bases  compare  ? 

12.  The  altitude  of  one  rectangle  is  10"  and  the  base  is 
4';  the  altitude  of  another  rectangle  is  20"  and  the  base  is  4'. 
What  is  the  ratio  of  their  areas  ?     Of  their  altitudes  ? 

13.  Two  rectangles  have  bases  20'  and  25'  and  both  have 
an  altitude  of  1 5'.  What  is  the  ratio  of  their  areas  ?  Of 
their  bases  ? 

14.  Two  rectangles  have  altitudes  of  8'  and  15'  and  both 
have  a  base  of  18'.  What  is  the  ratio  of  their  areas?  Of 
their  altitudes  ? 

15.  The  dimensions  of  one  rectangle  are  a  and  h  and  of 
another  are  a  and  c.  What  is  the  ratio  of  their  areas  ?  Of 
their  unequal  dimensions  ? 

1 6.  How  does  the  ratio  of  the  areas  of  the  rectangles  having 
equal  bases  compare  with  the  ratio  of  their  altitudes?  How 
does  the  ratio  of  the  areas  of  the  rectangles  having  equal  al- 
titudes compare  with  the  ratio  of  their  bases  ? 


Ratio,  Proportion  and  Similarity  151 

17.  Compare  the  ratio  of  the  areas  and  of  altitudes  of  tri- 
angles with  equal  bases.  Compare  the  ratios  of  areas  and  of 
bases  of  triangles  with  equal  altitudes.  Compare  the  ratios 
of  areas  and  of  altitudes  of  parallelograms  with  equal  bases. 
Compare  the  ratios  of  areas  and  of  bases  of  parallelograms 
with  equal  altitudes. 

Definition. — An  equation  of  ratios  is  called  a  proportion. 

For  example:    ^=\\    and  7=7-,    and  7=-  are  all  called 

proportions  and  are  sometimes  written  thus:  4  :  6  =  2  :  3, 
a  :  h=ac  :  be,  and  a  :  b=c  :  d.  The  last  is  read  "<i  is  to  h  as 
c  is  to  rf."  Read  the  other  two.  Numbers  that  form  a  pro- 
portion are  said  to  be  proportional. 

18.  Show  that  (i)  areas  of  rectangles  are  proportional  to 
their  bases,  if  their  altitudes  are  equal,  (2)  areas  of  rectangles 
and  their  altitudes  are  proportional,  if  their  bases  are  equal, 
(3)  areas  of  rectangles  are  proportional  to  the  products  of 
their  bases  and  altitudes,  (4)  areas  of  triangles  are  propor- 
tional to  the  products  of  their  bases  and  altitudes. 

19.  Is  2  :  5  =  6  :  15  or  |-=t\,  a  true  proportion?  Why? 
Is2:7=8:25a  true  proportion  ?  Give  reason  for  your  answer. 

Definition. — The  first  and  last  terms  of  a  proportion  are 
called  extremes;  the  second  and  third,  the  means. 

20.  Compare  the  product  of  the  extremes  with  the  prod- 
uct of  the  means  in  2  :  5  =  6  :  15;  in  3  :  7=6  :  14;  in  20  :  2 
=  10  :  i;  in  12  :  3=4  :  i.  What  do  you  find  true  of  the 
products  ? 

21.  For  which  of  the  following  expressions  is  the  product 
of  the  means  equal  to  the  product  of  the  extremes: 


(I)  i: 

:  3=4:  12  ; 

(6)     2 

:  3    =20  :  sa 

(2)  3 

:4  =  6:  12  ; 

(7)     8 

:  8a=3  :  3a; 

(3)  2: 

:  3=8:  11; 

(8)     oc: 

: y  =4^ : 4^ 

(4)  S: 

I  6  =  10  :  12  ; 

(9)  3«  : 

:  sx=6a  :  3^ 

(5)  8 

:3=i5  :6; 

(10)     X 

:sx=i  :3. 

152  First-Year  Mathematics 

Is  there  any  true  proportion  in  this  list  for  which  the  equaUty 
of  products  does  not  hold  ? 

Is  there  any  expression  in  the  list  that  is  not  a  true  pro- 
portion, for  which  the  product  of  the  first  and  fourth  num- 
bers is  equal  to  the  product  of  the  second  and  third  ? 

Principle. — In  any  proportion  the  product  of  the  means 
equals  the  product  of  the  extremes.  This  is  a  convenient  test 
of  proportionality.  , 

22.  By  this  test  tell  what  expressions  in  problem  21  are 
correct  proportions. 

23.  From  4X6=3X8,  write  a  proportion  in  as  many 
ways  as  you  can,  using  only  the  numbers  4,  6,  3,  and  8. 

24.  Show  by  using  arithmetical  numbers  that,  if  the  prod- 
uct of  two  numbers  (as  4X8)  equals  the  product  of  two  other 
numbers  (as  2X16),  either  pair,  that  is,  either  the  4  and 
8,  or  the  2  and  16,  may  be  made  the  extremes  and  the  other 
pair  the  means  of  a  proportion. 

25.  Make  a  proportion  in  at  least  four  ways  from  each  of 
the  following  equations,  and  show  by  the  law  for  testing  pro- 
portionaUty  that  your  proportions  are  true: 

(i)  2X10=4X5;  (6)  (a+by=mn; 

(2)4X9=3X12;  i'j)  m'—r'=a'-x'; 

(3)  .'=X8=4Xio;  (S)  x'—sx-\-2=a''+4a-s; 

(4)  3a=4&;  (g)  c''-d'=x'  +  2xy-\-y' ; 

(5)  ab=xy;  (10)  a'  —  c'=x'-r'. 

26.  Prove  that  if  a:b=b:c  then  b:a=c:b. 
Definition. — In  a  proportion  having  the  same  number 

for  both  the  second  and  third  te'rms,  this  number  is  called 
a  mean  proportional  between  the  other  two  terms.  Of  course, 
the  mean  proportional  may  be  made  the  first  and  fourth  terms 
of  a  proportion. 

27.  Show  that  the  mean  proportional  may  be  made  the 
first,  or  fourth,  term  of  a  correct  proportion  with  the  other  two 
numbers. 


Ratio,  Proportion  and  Similarity  153 

When  there  is  a  mean  proportional,  the  fourth  term  is  called 
a  third  proportional  to  the  other  two  numbers. 

28.*  Show  that  in  2  :  5  =  6  :  15  the  5  and  6  may  change 
places  giving  a  correct  proportion.  Show  also  that  the  2  and 
15  may  be  interchanged  without  destroying  the  proportionahty. 

Definition. — A  proportion  made  from  another  proportion 
by  interchanging  either  the  means  or  the  extremes  is  said  to 
be  formed  by  alternation.  In  the  same  way  the  first  propor- 
tion is  said  to  be  "taken  by  alternation.*' 

29.  Take  the  following  proportions  by  alternation  and  prove 
(by  the  test  principle)  that  you  obtain  true  proportions: 


(i)  8  :  11=24:  33  ; 

(6)  a  :  b=c  :  d  ; 

(2)  7  :  12=35  :  60; 

(7)  c:d=^x:  y  ; 

(3)  3  :  19  =  12  :  76; 

(8)  a  :  x=c  :  y  ; 

(4)  32  :  5  =  128  :  20; 

(9)  a  +  b:x=c+d:y; 

(5)  17:8  =  85:40; 

(10)  3a  +  i  :r=a  +  s  :  7 

§  73.     Similar  Figures  and  Proportionality 

1.  Draw  a  triangle  having  two  of  its  sides  10''  and  12" 
long  and  including  any  convenient  angle  between  them.  Call 
the  third  side  the  base.  Through  a  point  on  the  10"  side  and 
5''  from  the  vertex  of  the  included  angle  draw  a  parallel  to 
the  base.  Measure  the  distance  on  the  12"  side  from  the 
vertex  of  the  included  angle  to  the  crossing-point  of  the  parallel. 
How  does  the  ratio  of  the  parts  of  the  sides  cut  off  by  the  paral- 
lel compare  with  the  ratio  of  the  sides  10  and  12  ? 

2.  Draw  a  parallel  to  the  base  through  a  point  of  the  10'' 
side  2  J"  from  the  vertex  and  measure  the  distance  from  the 
vertex  to  the  crossing-point  of  the  2^  parallel  with  the  12" 
side.  Compare  the  ratio  of  the  parts  of  the  sides  with  the 
ratio  of  the  sides  themselves  (10  :  12). 

3.  Compare  the  ratios  of  the  corresponding  parts  of  the 
sides  made  by  parallels  to  the  base,  through  a  point  of  the 
10"  side  3''  from  the  vertex;  6"  from  the  vertex;  i\"  from 
the  vertex;   7^''  from  the  vertex;  8f  from  the  vertex. 


154  First-Year  Mathematics 

4.  Draw  two  triangles  having  the  same  shape  but  different 
sizes.  Measure  two  corresponding  pairs  of  sides  in  each  and 
compare  their  ratios.  Measure  another  pair  of  corresponding 
sides  and  compare  their  ratios.  What  seems  to  be  true  of  the 
ratios  of  corresponding  pairs  of  sides  of  triangles  having  the 
same  shape  (similar  triangles)  ? 

5.  Draw  two  squares  of  different  sizes  and  compare  the 
ratio  of  the  lengths  of  their  corresponding  sides. 

6.  Draw  two  rectangles  with  corresponding  sides  propor- 
tional but  differing  in  size  and  compare  the  shapes  of  the 
two  rectangles. 

Conclusion. — In  all  plane  figures  of  the  same  shape 
(similar  figures)  and  having  the  same  number  of  sides,  cor- 
responding sides  are  proportional. 

7.  Is  3:7=9:21  a  proportion?  Why?  Add  the  second 
term  to  the  first  and  the  fourth  to  the  third  and  ascertain 
whether  the  first  sum  is  to  seven  as  the  second  sum  is  to  21  ? 
Also  whether  the  first  sum  is  to  3  as  the  second  sum  is  to  9. 

8.  Make  two  triangles,  one  with  sides  3,  5,  and  7  and  the 
other  with  sides  9,  15,  and  21.  How  do  the  triangles  resemble 
each  other  in  shape,  or  form  ?  In  size  ?  How  does  the  ratio  of 
the  first  two  sides  of  the  first  triangle  compare  with  the  ratio 
of  the  two  corresponding  sides  of  the  second  triangle  ?  How 
does  the  ratio  of  any  pair  of  sides  of  one  of  the  triangles  com- 
pare with  the  ratio  of  the  corresponding  sides  of  the  other  ? 

9.  Show  by  drawing  triangles  and  extending  sides  what  is 
shown  with  arithmetical  numbers  in  problem  7  ? 

Definition. — When  a  proportion  is  made  from  a  given 
proportion  by  placing  the  sum  of  the  two  terms  of  each  ratio 
in  place  of  the  antecedent  (first  term)  of  each  ratio,  or  in  place 
of  the  consequent  (last  term)  of  each  ratio,  the  given  propor- 
tion is  said  to  be  taken  by  composition.  If  the  difference 
instead  of  the  sum  of  the  terms  of  each  ratio  is  used  in  this 
way,  the  given  proportion  is  said  to  be  taken  by  division. 


Ratio,  Proportion  and' Similarity  155 

10.  Write   a   proportion,  first,   by   composition,   then   by 

division  from  the  following,  and  prove  that  the  proportions 
you  write  are  correct: 

(1)8:3=16:6;  {6)  a-h:b=c-d:d\ 

(2)  12  :  10=36  :  30  ;  (7)  0+3  :  a=z  +  6  :  6  ; 

(3)  5:2   =30:  12;  (8)  a:h=c:d; 

(4)  15  :  12  =  10  :  8  ;  (9)  ^  •  c=^  :  2  ; 

(5)  4a  :  a=86  :  6  ;  (10)  x  :a=c  :d  . 

§  74.     Problems  and  Applications 

Find  the  value  of  the  letter  in  each  of  the  following  pro- 
portions : 


I. 

X 

2  =  12  :  I . 

6. 

5-:^;:  3  =  12  :  2. 

2. 

8 

x  =  2/^  :  2  . 

7- 

a-|-8  :  9=36  :6. 

3- 

7 

35=^^:  25. 

8. 

iS:35;-t-i=6:3  . 

4. 

9 

11=54:^. 

9- 

28  :  7=4:x;— 2  :  14 

5- 

18 

3=z: I. 

10. 

^-5:5=4:9- 

11.  If  17  acres  of  land  cost  $850  what  would  51  acres  cost, 
at  the  same  price  ? 

12.  The  force  /  required  to  hold  a- load,  L,  from  rolhng 
down  a  slope,  as  AC,  is  found  by  the  proportion  /:L  =  BC:  AC; 
that  is,  the  force  is  to  the  load  as  the  height  BC  of  the  slope 
is  to  its  length  AC.  Find  the  force  that  is  needed  to  prevent 
a  load  of  2,000  lb.  from  rolling  down  a  slope  that  is  1,200  ft. 
long  and  100  ft.  high. 

13.  Find  the  force  needed  to  prevent  roUing  downward 
of  the  following  loads,  the  lengths  and  heights  of  the  slope, 
as  indicated : 

(i)  Z,  =  i,5oo,        AC  =  2,oooft.,     BC  =  2ooft. 

(2)  L  =  2,8oo,        AC=3,563ft.,    BC  =  509  ft. 

(3)  1,= 5,000,        AC  =  I  mi.,         BC  =  528ft. 

(4)  Z,=3o  tons,     AC  =  2  mi.,         BC  =  528  ft. 

(5)  L  =  iSor,         AC  =  I  mi.,         BC  =  i58.4ft.; 

(6)  L=3oor,         AC  =  imi.,         BC  =  792  feet. 


156  First-Year  Mathematics 

14.  The  weight  of  sheet  metal  is  proportional  to  the  area 
of  the  sheet.  If  a  rectangle  of  sheet  zinc  18  in.X24  in.  weighs 
7i  oz.,  how  much  does  a  rectangle  24  ft.XiS  ft.  of  the  same 
sheet  weigh  ? 

15.  Fifteen  sq.  ft.  of  sheet  copper  weighs  2^  lb.  How 
much  will  a  piece  of  the  same  sheet  8  ft.  by  24  ft.  weigh  ? 

16.  If  ^i  denote  the  time  it  takes  for  a  pendulum  of  length 
Lj  to  vibrate  once,  and  /,  and  L^  denote  similar  numbers  for 
another  pendulum,  a  law  of  vibration  is  (i)  /i :  ^3  =  y^xT  •  l/zT 
which  gives  (2)  t\  :  tl=Lj^  :  Lj.  If  a  pendulum  36  in.  long 
vibrates  in  .5  sec,  how  long  will  it  take  a  pendulum  49  in. 
long  to  vibrate?     25  in.  long?     64  in.  long?    81  in.  long? 

17.  If  a  pendulum  39  in.  long  vibrates  in  i  second,  how 
long  will  it  take  a  pendulum  13  ft.  long  to  vibrate  ? 

18.  State  in  words  the  laws  of  problem  15,  (i)  and  (2), 
for  vibrating  pendulums. 

19.  The  areas  of  similar  triangles  are  as  the  squares  of 
corresponding  sides.  Show  by  means  of  a  figure  that  this 
is  true. 

20.  The  side  of  a  triangle  is  8  in.  long  and  its  area  is  24 
sq.  in.  The  corresponding  side  of  a  similar  triangle  is  12  in. 
What  is   the  area  of  the  second  triangle  ? 

21.  The  areas  of  all  similar  figures  are  proportional  to 
the  squares  of  corresponding  sides.  Show  by  a  figure  what 
this  means. 

22.  If  two  parallelograms  are  similar,  and  a  side  and  the 
area  of  one  are  6  ft.  and  45  sq.  ft.,  respectively,  and  the  side 
of  the  second  parallelogram  which  corresponds  to  the  6  ft. 
side  is  3  ft.,  what  is  the  area  of  the  second  parallelogram  ? 

23.  The  volumes  of  similar  solids  (solids  of  same  shape, 
corresponding  edges  proportional)  are  as  the  cubes  of  corre- 
sponding edges,  or  of  corresponding  dimensions.  Show  by  a 
sketch  what  this  means. 

24.  If  a  watermelon  6  in.  across  costs  15  cents,  what  would 


Ratio,  Proportion  and  Similarity  157 

another  cost  at  the  same  rate,  whose  dimension  correspond- 
ing to  the  6  in.  line  is  10  inches  ? 

25.  A  box  one  of  whose  edges  is  18  in.  holds  3  bu.  What 
would  a  box  of  the  same  shape  hold,  if  the  corresponding 
edge  were  63  inches  long  ?     2^  inches  long  ? 


CHAPTER  XXI 


LINEAR  EQUATIONS  CONTAINING  TWO  UNKNOWN  NUMBERS 
§  75.     Plotting  Linear  Equations 

1.  The  sum  of  two  numbers  is  4,  their  difference  is  2. 
Find  the  numbers. 

2.  The  sum  of  two  numbers  is  5.     Find  the  numbers. 
Solution:    The  equation  is  ^+>'=5  , 

Let  X  be  some  number,  as  i,    then  y=4; 
if  x=2  ;  if  :x;=5  ;  if  x=6^  ; 

then  y  =  T,  ;      then  y=o\      then  y=  —  i\,  etc. 


X 

y 

I 

2 

5 
6i 

4 

3 
0 

Assume  five  other  values  oi  x  and  compute  the  correspond- 
ing values  of  y.     Tabulate  as  above. 

It  is  seen  that,  if  a  linear  equation  contains  two  unknown 
numbers,  for  every  value  of  one  of  the  unknown  numbers  there 
is  a  value  of  the  other.  Every  pair  of  numbers  in  the  tables 
satisfies  the  given  equation.  Hence,  there  is  an  indefinitely 
large  number  of  solutions  of  a  single  eqimtion  containing  two 
unknown  numbers. 

3.  Find  the  pair  of  numbers  which  satisfies  both  equations 
of: 

x^   y=   i,,  (3)  3X-2:x;=-4, 

2^+3^=12;  2X—  y=  —  i; 

Sx+2y=  i;  (4)  2,x+2y=26, 

?,x—  y  =  io\  5^+9>'=83. 

158 


(I) 
(2) 


Linear  Equations  Containing  Two  Unknown  Numbers  159 

§76.     Graphical  Solution  of  Simultaneous  Equations  of 
the  First  Degree 

I.  Given    (i)  x-\-2y=ii , 
and  (2)  2X-\-y=i2,  . 

Assign  to  X  any  three  or  more  values  and  from  each  equa- 
tion find  the  corresponding  value  of  y,  thus  getting  several 
solutions.  Tabulate  the  results.  On  squared  paper  find 
the  graphs  of  the  equations  of  (i)  and  (2). 

Do  the  co-ordinates  of  points  on  the  graph  of  equation  (i) 
satisfy  equation  (i)  ?    Equation  (2)  ? 

Note. — The  x-  and  y-distances  are  the  co-ordinates. 

Do  the  co-ordinates  of  points  on  the  graph  of  equation  (2) 
satisfy  equation  (2)?  Equation  (i)  ?  From  the  graphs  find 
the  solution  which  satisfies  both  equations  (i)  and  (2). 

Solve  in  a  similar  way  by  graph  the  following  pairs: 

2.  2,x-Ay=  1,         3-  2x-l-3>'  =  i9,         4-  3^-f>'=  3> 
x-\-2y=  9.  3X-|-2^  =  i6.  2X         _ 

~-^y-  4. 

§  77.     Equivalent  Equations 

1.  Draw  graphs  of  x—y==2, 

and  2X=2y-\-A,. 

Is  there  a  common  solution  of  these  equations  ?  Two 
linear  equations  with  two  unknowns  having  all  of  their  solu- 
tions in  common  are  called  equivalent  equations.  How  can 
one  of  these  equations  be  derived  from  the  other  ? 

2.  Are  the  equations    2:x;— ^  =  3  ? 

and  6x = 9  -}-  3  ;y  equivalent  ? 

Answer  the  latter  question  both  graphically  and  algebraically. 

3.  Make  up  two  equivalent  equations.  Prove  them  equiva- 
lent. 


i6o  First-Year  Mathematics 

§  78.     Inconsistent  Equations 
2X+  y=6, 


1.  Draw    graphs    of    , 
(  4x+2y=4. 

Is  there  a  common  solution  for  these  equations  ? 

Two  Unear  equations  with  two  unknowns  having  no  com- 
mon solution  are  called  inconsistent  equations. 

2.  Show  by  graph  that  $x—y=io 

and        $x=2-{-y  are  inconsistent. 

§  79.     Elimination 

I.  Solve   ^x-\-4y=22 

and  3:x;+  y=  9,  for  x  and  y. 
To  eHminate  x  (i.  e.,  to  remove  x),  multiply  the  first  equation 
through  by  3,  and  the  second  by  5.     The  resulting  equations: 

i5x-{-i2y  =  66 , 

isx-\-  sy=45, 

are  equivalent,  respectively,  to  the  two  given  equations.  There- 
fore, they  have  the  same  solution  as  the  two  given  equations. 
Subtracting  member  from  member,  there  results 

7y=2i, 
or  y=s- 

The  value  of  x  may  be  found  from  the  given  equations  by 
multiplying  the  second  of  the  given  equations  through  by  4, 
and  subtracting  as  before.      Do  this. 

Or,  we  may  find  the  value  of  x  by  replacing,  in  the  second 
of  the  given  equations,  y  by  its  value,  3,  which  gives: 

3^+3=9 » 
whence  3r!c=6, 

and  x=2 . 


Linear  Equations  Containing  Two  Unknown  Numbers  i6i 

2.  Solve  by  elimination  the  following  simultaneous  equa- 
tions : 

jsx  +  2y  =  i2,  {ax+by  =  i  , 

I  4x-sy=-i  ;  /  bx-{-ay  =  i  ; 

.  .    (  %a  —  2b  =  i  ,        '  /  I  ,  2 


8a  =  56-ii  ;  (8)  <  ^    ^ 

2a     b  I  2_2_    . 

^'^''  ^  a     26  /     I      ,      I 

-H —  =  20;  1— 1 — —  =  1, 

23  U-{-x     i  +  y 

j  ii/-io7;  =  i4,  /     2 L-  =  i. 

(4)"{    5^+   7^=41;  \i+^     i  +  y     '' 

-5^^        _  /5^-2_  5y+7 

-8-  +  7^-^^'  (10)     3^-^     ^^+^^' 

^^^  ^    iiv    St         _  I  ^-i  ^2^-5 


(6) 


12      8     '^'  ^3^+5     6^+3 

^+3-53'=6; 


3.  The  rainfall  in  a  certain  locality  one  year  was  ^^-  as 
much  as  it  was  the  year  after;  and  the  total  rainfall  for  the 
two  years  was  18.5  inches.  Determine  the  amount  for  each 
year? 

4.  A  man  who  can  row  6  miles  an  hour  down  stream  can 
row  two  miles  an  hour  up  stream.  What  is  the  speed  of  the 
current  ?     What  is  the  speed  of  rowing  in  still  water  ? 

5.  By  means  of  a  chain  running  over  the  sprocket  wheels 
of  a  bicycle  the  small  wheel  is  driven  by  the  larger  one.  The 
small  wheel  makes  90  revolutions  a  minute  more  than  the 
large  one.  If  the  small  wheel  were  \  smaller  than  it  is,  it 
would  make  120  revolutions  more  than  the  large  one.  How 
many  revolutions  per  minute  does  each  sprocket  wheel 
make? 


i62  First-Year  Mathematics 

6.  A  man  and  a  boy  together  receive  a  wage  of  $37.50. 
The  man  works  10  days  and  the  boy  8  days.  The  man  earns 
in  4  days  $3 .  50  more  than  the  boy  earns  in  6  days.  What 
amount  does  each  receive  ? 

7.  If  the  altitude  of  a  rectangle  be  increased  by  4  inches 
and  its  base  be  diminished  by  2  inches  the  area  will  be  increased 
by  22  square  inches.  If  the  altitude  be  increased  by  i  inch 
and  the  base  be  diminished  by  i  inch,  the  area  will  be  increased 
2  square  inches.  Find  the  base  and  altitude  of  the  given 
rectangle. 

8.  A  bar  of  iron  of  uniform  thickness  10  ft.  long  and 
weighing  \\  cwt.  is  supported  at  its  extremities  in  a  horizontal 
position,  and  carries  a  weight  of  4  cwt.  suspended  from  a 
point  3  feet  from  one  extremity.  Find  the  pressure  on  the 
points  of  support.  (Take  the  turning-point  at  the  middle 
point  of  the  beam.) 

9.  A  bar,  weighing  7  lb.  per  linear  foot  rests  on  a  ful- 
crum three  feet  from  one  end.  What  must  be  its  length,  that 
a  weight  of  71^  lb.  suspended  from  that  end  may  just  be 
balanced  by  20  lb.  suspended  from  the  other  ?  What  is  the 
pressure  at  the  fulcrum  ?  (Take  the  turning-point  at  the 
midpoint.) 

10.  A  beam  of  uniform  thickness  weighing  10  lb.  is  sup- 
ported at  the  ends  by  the  two  props.  A  weight  of  31  lb.  hangs 
4  ft.  from  one  end.  Find  the  length  of  the  beam  and  where 
a  weight  must  be  placed  that  the  pressure  on  the  two  props 
may  be  i8  and  25  pounds. 

1 1 .  A  signboard  four  feet  long  hangs  in  a  horizontal  posi- 
tion by  hooks  at  its  ends  to  an  iron  rod  weighing  14  lb.  The 
rod  is  supported  at  its  ends.  If  the  signboard  weighs  32 
lb.  how  far  from  the  ends  of  the  rod  must  the  hooks  be  placed 
that  the  pressure  at  the  ends  of  the  rod  may  be  12  lb.  and 
34  lb.? 


Linear  Equations  Containing  Two  Unknown  Numbers  i6^ 

12.  Write  out  the  equations,  from  the  laws  of  force,  and 
find,  for  each  of  the  six  problems  in  the  table  below,  the  force, 
or  arm  not  given  in  the  table. 


dr 

Fr 

d. 

F, 

d. 

Fs 

I 
II 

8 

-  S 

8 

o 

6 

—  lO 

2 

2 

-5 

III 

4 

2 

3 

9 

-7 

IV 

—  2 

II 

8 

-3 

6 

V 

8 

—  2 

2 

2 

3 

VI 

7 

5 

2 

-5 

-5 

CHAPTER  XXII 

QUADRATIC  EQUATIONS 

§  80.     Uses  of  Quadratics  and  Interpretation  of  Negative  Roots 

1.  A  body  is  thrown  vertically  upward  with  a  speed  of 
w(  =  5o)  feet  per  second.  It  passes  a  point  d(  =  2^)  feet  from 
ground  twice,  first,  going  upward,  and  second,  on  its  return. 
Its  speed  v  at  this  point  can  be  found  from  the  equation 

v'=u'  —  2gd  =  $oXso  —  64  ■  25  , 
Hence  1^^=900. 

Extracting  the  square  roots  of  both  sides  of  this  equation 
gives  '^  =  30  and  7;=— 30.  The  first  being  the  speed  of  the 
particle  when  moving  in  the  direction  in  which  it  was  thrown, 
the  second  its  speed  when  moving  in  the  opposite  direction 
on  its  return. 

2.  Find  the  value  of  x  in  x^=4. 

Extracting  the  square  root  of  both  sides  of  the  equation, 
gives  :v=2,  or  :v=— 2. 

Check  by  substituting  each  one  of  these  values  in  the 
original  equation. 

3.  Solve  for  x: 

x^ 
(i)  x'  =  62s;  (4)  -=5; 


(2)  :x;='-i2i=o;  (5)  ^=14- 


(3)  3^'-27oo=o; 
4.  If  a  particle  is  dropped  so  that  it  can  fall  freely,  the 
attraction  of  the  earth  increases  its  speed  by  32  feet  per  second. 
This  increase  is  denoted  by  the  letter  g.  If  in  /  seconds  the 
particle  has  moved  5  feet,  then  s  =  ^gt^.  Its  speed  v  is  found 
from  the  equation  v'  =  2gs. 

164 


Quadratic  Equations  165 

(i)  How  long  will  it  take  a  ball  to  fall  2,304  feet? 

(2)  How  long  will  it  take  a  body  to  fall  1,600  feet? 

(4)  What  would  be  the  velocity  with  which  the  body  in 
the  last  problem  would  strike  the  earth  ? 

(4)  A  body  falls  from  rest.  How  long  must  it  move  to 
acquire  a  velocity  of  836  feet  a  second  ? 

5.  If  a  force  of  P  pounds  acting  on  a  body  of  weight  W 
moved  it  through  a  distance  of  s  feet,  the  velocity  acquired 

can  be  found  from  the  equation:     Ps= . 

A  hammer,  weighing  16  lb.,  strikes  a  post  with  a  force  of 
480  lb.,  and  drives  it  ^^  of  an  inch.  Find  the  velocity  with 
which  the  hammer  moved  at  the  instant  it  strikes  the  post. 

6.  The  law  of  vibration  of  a  pendulum  is  gt^=Tr^l.  In 
what  time  will  a  pendulum  vibrate  whose  length  is  1 5  inches  ? 

§81.     Formal  Problems 

Solve  the  following  equations: 

I.  (:v+4)^=9;  2.  {x+ky=g. 

In  exercises  3-1 1,  add  something  to  both  sides  that  will  make 
the  first  member  of  the  following  equations  a  perfect  trinomial 
square  and  find  the  values  of  the  unknown  : 

3.  x''-\-2X=9>\  7.  3'=*— 3^  +  1=0  ; 

4.  r)£;^+6x=— 5;  8.  2^2— 3»— 2=0; 

5.  x^-\-2ax=i6—a^  ;  9.  5r*  +  ii=3^; 

6.  x^+4^— 1=0;  10.  6y'^-\-e)ay=—a\ 


II.  Solve  ax^+hx-'rc=o.     Result,  —   " 


2a 


Definition. — A  more  scientific  meaning  of  the  phrase  to 
"  solve  an  equation  "  is  to  find  the  root,  or  the  roots,  of  the  equa- 
tion. A  root  of  an  equation  is  the  value,  or  values,  of  the 
unknowns  in  terms  of  the  coefficients. 


1 66  First- Year  Mathematics 

12.  Solve  the  following  by  use  of  the  formula  in  Problem  1 1 : 

(i)  x^  —  $x+6=o  ;  (6)  3M'— 4W  — io=o; 

(2)  25c='  +  9  =  9X  ;  (7)  245;— 3c*=o; 

(3)  x'-\-22(x-\-s)=o  ;  (8)  6x'-sx  =  6; 

(4)  2i+:v;=2a:^;  (9)  7/^-37/= -10  ; 

(5)  mx'  +  nx-\-p=o  ;  (to)  25*  — 135=  — 20. 

13.  Solve  by  factoring: 

(i)  x'  —  2X=i^  ;  (6)  (x—i)(x+i)(x—2)=o; 

(2)  :)c^  — 5x+6=o  ;  (7)  x^+x'—x—i=o; 

(3)  6x*— :»— 1=0;  (8)  :x:^  +  o:!C+6:x;+fl&=o  ; 
(4)2^^-7^-15=0;  (9)  6y'-s2y=-s2; 

,  (5)  :»(:x;+7)  =  7(^+28);    (10)  6^'  +  5/»=6. 

§  82.     Problems  Leading  to  Quadratics 

In  the  following  obtain  two  different  expressions  for  some 
number,  or  magnitude;  write  the  expressions  equal  to  each 
other,  and  solve  the  resulting  equation.  Finally,  test  the  cor- 
rectness of  your  results  by  substituting  in  the  equation  first 
formed. 

1.  One  side  of  a  rectangle  is  3  ft.  longer  than  the  other. 
If  the  longer  side  be  diminished  by  one  foot  and  the  other 
side  increased  by  one  foot  the  area  of  the  rectangle  will  be 
30  sq.  ft.     How  long  is  the  rectangle  ? 

2.  The  area  of  a  rectangle  is  54  sq.  in.  and  the  sum  of  its 
length  and  breadth  is  1 5  inches.     How  long  is  the  rectangle  ? 

3.  Find  the  length  of  a  rectangle  whose  area  is  60  sq.  in. 
and  the  sum  of  whose  breadth  and  length  is  16  inches. 

4.  The  diagonal  and  the  longer  side  of  a  rectangle  are 
together  five  times  the  shorter  side,  and  the  longer  side  exceeds 
the  shorter  by  33  yards.     What  is  the  area  of  the  rectangle  ? 

Principle. — In  a  right  triangle  the  square  on  the  hypoth- 
enuse  equals  the  sum  0}  the  squares  on  the  other  two  sides. 


Quadratic  Equations  167 

5.  The  hypothenuse  of  a  right  triangle  is  5  in.  and  one  of 
the  sides  is  i  inch  longer  than  the  other.  Find  the  length  of 
the  sides. 

6.  Two  trains  100  miles  apart  on  roads  which  cross  at 
right  angles  are  running  toward  the  crossing.  One  train 
runs  10  mi.  an  hour  faster  than  the  other.  At  what  rates  must 
they  run,  if  they  both  reach  the  crossing  in  two  hours  ? 

7.  A  tree  was  broken  by  a  storm  so  that  the  top  touched 
the  ground  50  ft.  from  the  foot  of  the  tree.  The  stump  was 
I  of  the  height  of  the  tree.     What  was  the  height  of  the  tree  ? 

8.  Two  trains  move  from  a  crossing  in  perpendicular  direc- 
tions. One  has  a  speed  of  12  mi.  per  hr.,  the  other  of  16  mi. 
per  hour.     In  how  many  hours  will  they  be  70  mi.  apart  ? 

9.  If  V  is  the  velocity,  /  the  time,  5  the  distance  passed 
over  by  a  body  thrown  vertically  downward,  then  (i)  v  =  u-\-gt 
and  (2)  s=ut+\gt^,  where  u  is  the  velocity  the  body  has  at 
the  start  {initial  velocity).  If  the  body  is  thrown  vertically 
upward,  the  following  laws  apply:    (i)  v  =  u—gt;    (2)  s  =  Ht 

-i^gt^;   and  (3)  5  =  —--. 

(i)  With  what  initial  velocity  must  a  bullet  be  fired  up- 
ward that  it  may  rise  to  a  height  of  6,400  feet  ? 

(2)  A  body  is  projected  upward  with  a  velocity  of  80  ft. 
per  second;  in  what  time  will  it  return  to  hand?  Answer: 
In  5  seconds. 

(3)  A  falling  body  starts  with  a  velocity  of  30  ft.  a  second. 
When  will  it  have  gone  just  300  feet  ? 

(4)  A  ball  is  thrown  vertically  into  the  air  with  a  velocity 
of  40  ft.  a  second.     When  will  it  be  at  the  height  of  1 6  feet  ? 

(5)  How  hard  must  I  throw  a  ball  that  it  shall  just  reach  a 
man  on  a  scaffold  25  ft.  above  me  ? 


CHAPTER  XXIII 

LOGARITHMS 
§  83.     Logarithms  of  Exact  Powers  of  10 

Note. — The  teacher  is  to  work  this  chapter  through  with 
the  class.  Review  here  the  ideas  "factor,"  "power,"  "root," 
and  "exponent." 

Express  100  as  factors  in  10;  thus,  100  =  10X10.  Express 
1,000  similarly;   10,000;   100,000. 

-  Express  100,  1,000,  10,000,  and  100,000  as  powers  of  10; 
thus  100  =  10^,  etc. 

What  has  been  done  may  be  shown  thus: 


The  Number 

Expressed  as  Factors 

Expressed  as 
Powers  of  10 

100 

10X10 

10=" 

1,000 

loXioXio 

I03 

10,000 

loXioXioXio 

I04 

100,000 

loXioXioX  10X10 

I05 

What  does  the  small  figure,  placed  at  the  right  and  above 
the  lo's  in  the  last  column  express? 

This  number  written  in  small  figures  to  the  right  of  and 
above  the  10,  is  called  the  exponent  of  the  power. 

Distinguish  between  the  power  and  the  exponent  in  each 
of  the  following  numbers:  2*,  2^,  2^,  3^,  33^  4^^  43^  53^  a',  a^,  x^. 

Definition. — The  number  (exponent)  which  expresses 
how  many  times  10  must  be  used  as  a  factor  to  give  a  certain 
number  is  called  the  logarithm  of  the  latter  number.  Thus, 
in  i,ooo  =  io3,  the  3  is  the  logarithm  of  1,000.  Give  other 
examples  from  the  foregoing  table.  Give  the  logarithm  of 
1,000,000;  of  I  with  7  zeros  following;  of  i  with  12  zeros 
following,  etc. 

Fill  out  the  logarithm  column  of  the  following  table: 
168 


Logarithms  169 

The  Number  Power  Logarithm 

100 10* 2 

1,000 lo3 

10,000 lO'* 

100,000 lo5 

1 ,000,000 lO*^ 

Starting  with  the  fourth  number  of  column  i,  how  may 
the  third  number  of  the  same  column  be  obtained  from  it? 
The  first  from  the  second  number  ?  What  number,  following 
the  same  law,  could  be  added  above  100  in  column  i  ?  Write 
it  in  its  proper  place. 

What  second  number  in  column  i  could  be  added  above 

the  10,  by  following  the  same  law  ?    Write  it  in  column  i  also. 

State  in  words  the  law  of  the  numbers  of  column  i. 

Law  stated. — Any  number  0}  column  i  may  he  obtained 

from  the  number  next  below  it  by  dividing  the  number  next 

below  by  10. 

State  the  same  law  using  the  word  ''multiplying"  instead 
of  "dividing,"  making  such  other  changes  in  the  wording  as 
are  necessary. 

Starting  with  the  last  number  of  column  3,  how  is  the  next 
to  the  last  obtained  from  it  ?  The  second  from  the  third 
number  ?    The  first  from  the  second  ? 

Add  another  number  above  the  2,  following  the  law  of  the 

numbers  of  column  3.     Add  a  second  number  above  this  "i." 

State  the  law   by  which   each  number   of   column   3    is 

obtained  from  the  number  next  below  it;  from  the  number 

next  above  it. 

What  then  is  the  logarithm  of  10  ?     Of  i  ? 
Arrange  what  has  been  found  thus: 

Nximber  Power  Logarithm     , 

I 10° o 

10 10^ I 

100 10' 2 

1,000 I03 3 

10,000. lO'* 4 

100,000 I05 5 


170  First-Year  Mathematics 

If  a  number  lies  between  10  and  100,  between  what  two 
numbers  must  its  logarithm  lie?  Between  i  and  10?  100 
and  1,000?     1,000  and  10,000? 

What  kind  of  numbers  lie  between  o  and  i  ?  Answer: 
Fractions  less  than  i .  Between  i  and  2  ?  Between  2  and  3  ? 
3  and  4  ?  4  and  5  ?  Give  examples  of  numbers  lying  in  each 
of  the  intervals  just  mentioned. 

Give  examples  of  numbers  that  lie  between  i  and  10? 
10  and  100?  100  and  1,000?  1,000  and  10,000?  How 
may  numbers  that  lie  in  each  of  the  last-mentioned  inter- 
vals be  recognized  ? 

The  logarithm  of  a  certain  number  is  1.684.  Between 
what  two  numbers  of  column  i  of  the  table  does  this  num- 
ber lie  ? 

State  between  what  two  numbers  of  column  i  numbers 
having  the  following  logarithms  lie:    (i)   2.672;    (2)    .486; 

(3)  3125;    (4)  4061;    (5)    .186;    (6)    .006. 

Between  what  two  numbers  of  column  3  must  the  logarithms 
of  the  following  numbers  lie:    (i)  18;    (2)  7.84;    (3)  25.74; 

(4)  3.62;    (5)  268.7;    (6)  1,286;    (7)  68,491? 

The  question  now  arises  how  may  the  logarithms  of  num- 
bers that  are  not  exact  powers  of  10  be  obtained  ? 

§  84.     Logarithms  of  Numbers  that  are  not  Exact  Powers  of  10 

How  may  the  next  to  the  last  number  (the  4)  of  column  3 
of  the  last  table  above  be  found  from  the  last  (the  5)  ? 

How  may  the  next  preceding  (i.  e.,  the  3)  be  found  from 
the  next  to  the  last  (the  4)  ? 

How  may  each  number  of  column  3  be  found  from  the 
number  last  below  it  ?    From  the  number  next  above  it  ? 

What,  then,  is  the  law  of  the  numbers  of  column  3  ? 

Answer:  Each  number  of  column  5  may  be  obtained  by 
adding  a  constant  number  (i.  e.,  i)  to  the  number  next 
above  it. 


Logarithms  171 

State  the  same  law  using  the  word  "subtracting"  for 
"adding,"  making  other  necessary  changes  in  the  wording. 

How  then  might  another  number  be  put  in  between  each 
pair  of  numbers  of  column  3  following  a  similar  law  ? 

Answer:  By  adding  .5  to  each  number  of  column  3  ex- 
cepting the  last.     We  should  obtain  thus:    o,   .5,  i,  1.5,  2, 

2-5.  3,  3-5>4,  4-5.  5- 

In  a  similar  way,  we  could  obtain,  by  putting  other  num- 
bers between  these:  o,  .25,  .50,  .75,  i,  1.25,  1.50,  1.75,  2, 
2.25,  2.50,  2.75,  3,  3.25,  3.50,  3.75,  4,  4.25,  4.50,  4.75,  5. 

Let  us  now  see  whether  we  may  insert  other  numbers 
between  those  of  column  i  in  accordance  with  the  law  of- the 
numbers  already  in  column  i ,  with  the  aid  of  a  factor  different 
from  10. 

To  this  end  let  us  write  down  the  law  of  the  numbers 
already  in  column  i.  How  may  each  number  of  the  column 
be  obtained  from  the  number  next  below  it?  The  next 
above  it  ? 

What  is  the  law  of  the  numbers  of  column  i  ? 

Answer:  Each  may  be  obtained  by  multiplying  the  one 
next  above  it  by  a  certain  number  (i.  e.,  10),  or  by  dividing  the 
number  next  below  it  by  this  same  number  (10). 

For  the  new  series  of  numbers  we  wish  to  obtain  by  put- 
ting other  numbers  between  those  of  column  i,  let  the  con- 
stant multiplier  (or  constant  divisor)  be  x.  For  the  number 
between  10  and  100  we  should  have  loar,  and  multiplying  this 
10^  again  by  x,  the  product  must  then  be  100.    (why  ?) 

Or  we  have: 

I  ox'  =  1 00,  whence  ^e'  =  i  o.     (Why  ?) 
This  requires  the  solution  of  a  quadratic  equation. 

Extracting  the  square  root  of  10,  we  find  :x;=3.i623 
(omitting  the  negative  result,  —3.1623). 

We  now  have  the  new  series:  i,  3.16,  10,  31.62,  100, 
316.23,  1,000,  3,162.3,  10,000. 


172  First-Year  Mathematics 

How  are  31 .62,  316.23,  and  3,162.3  found  ? 

We  may  again  fit  numbers  between  these  by  taking  our 
multiplier  y,  and  noticing  that  between  10  and  31.62,  for 
example,  we  should  have  loy  and  also,  that  ioy'=^i.62,  or 
that  j^  =3. 1 623  (why?). 

Whence,  extracting  the  square  root  of  3.1623,  we  find 
^=1.7785  (again  omitting  the  —1.7785). 

This  gives  the  series:  i,  i  .78,  3.16,  5.62,  10,  17.78,  31 .62, 
56.23,  100,  177.85,  316.23,  562.34,  1,000,  1778.50,  3,162.30, 
5,623.40,  10,000. 

How  are  5.62,  17.78,  56.23,  177.85,  316.23,  etc.,  found? 

Arranging  these  numbers  in  a  table,  with  a  third  column 
to  show  their  meanings,  we  have: 


Numbers 

Logarithms 

Meanings 

1 .00 

0.00 

1 .00  =  10-°'' 

1.78 

0.25 

l.78  =  lo-«S 

3.16 

0.50 

3.i6  =  io-so 

5.62 

0-75 

5.62  =  10-75 

10.00  • 

1 .00 

io.oo=io*-<»*' 

17.78 

1-25 

i7.78  =  io^-»s 

31.62 

1.50 

31 .62  =  10^-50 

56.23 

1-75 

56.23  =  10*-" 

100.00 

2.00 

ioo.oo  =  io'-<» 

177-85 

2.25 

i77.85  =  io»-»s 

316.23 

2.50 

3i6.23  =  io'-so 

562.34 

2.75 

562.34  =  io»-w 

1 ,000 .  00 

3.00 

1,000. 00  =  I03-«> 

1,778.50 

3-25 

1,778. 5o  =  io3-«s 

3,162.30 

3-50 

3,162.30  =  103-50 

5*623.40 

3-75 

5,623.40  =  103-75 

10,000.00 

4.00 

10,000.  00  =  104-«» 

Evidently  other  numbers  might  be  inserted  in  columns  i 
and  2  and  the  table  extended  indefinitely.     Such  extensions 


Logarithms 


173 


have  been  made  by  computers,  and  the  numbers  together 
with  their  logarithms  have  been  arranged  in  tables  convenient 
for  use.  It  is  obvious  that  such  a  table  could  be  extended 
either  downward,  or  by  the  insertion  of  other  numbers  between 
those  given  in  the  table.  Carrying  the  extension  one  step 
farther,  we  have  the  following  table  of  numbers  and  their 
logarithms. 


Number 
I .00000. 
I .07461. 
1. 15478. 
I .24094. 

I -33352  • 
1.43302. 

I -53993 • 
1.65482. 
1.77828. 
1.91205. 
2.05352. 
2.20673. 

2-37137- 
2.54830. 
2.73842. 
2.94263. 
3.16229. 
3.39821. 

3-65174- 
3.92419. 
4.21695. 

4-53158- 
4.86968. 

5-23299- 


Logarithm  Number 

. 00000  5 . 62340 

.03125       6.04296 

.06250       6.49382 

-09375       6.97830 

.12500       7.49894 

.15625       8.05842. 

.18750       8.65964 

.21875       9-30572 

.25000      10.00000 I 

.28125      10.74610 I 

.31250      11.54780 I 

•  34375     12.40940 I 

-37500     13-33520 I 

.40625     14.33020 I 

•43750     15-39930 1 

.46875     16.54820 I 

.50000     17.78280 I 

.53125     19.12050 I 

.56250     20.53520 I 

.59375     22.06730 I 

.62500     23.71370 I 

.65625     25.48300 I 

.68750     27.38420 I 

.71875     29.42630 I 


Logarithm 
.75000 
.78125 
.81250 

•84375 
•87500 
.90625 

•93750 
.96875 
I . 00000 
•03125 
.06250 

■09375 
.12500 

.15625 
.18750 
-21875 
.25000 
.28125 
-31250 

•34375 
•37500 
.40625 

-43750 
•46875 


The  tables  to  be  used,  called  Four-Place  Tables  by  Beebe, 
contain  the  fractional  parts  (called  mantissas)  of  the  logarithms 


174  First-Year  Mathematics 

to  four  places  of  decimals  of  the  numbers  i  to  10,000.  Mem- 
bers of  the  class  will  provide  themselves  with  a  copy  of  these 
tables,  or  other  four-place  tables. 

§  85.     Meaning  and  Use  of  a  Four-Place  Table  of  Logarithms 

A  four-place  table  of  common  logarithms  contains  the 
fractional  parts  (the  mantissas)  of  the  logarithms  of  all  num- 
bers from  10  to  1,000.  The  integral  part  (called  the  charac- 
teristic) of  the  logarithm  is  easily  supphed  by  recalUng  that 
all  numbers  between  i  and  10  have  one  digit  on  the  left  of 
the  decimal  point,  and  that  their  logarithms  are  all  o-f  a 
fraction  less  than  i,  etc. 

Give  examples  from  a  table  of  four-place  logarithms. 

§  86.     To  Find  from  a  Table  the  Logarithm  of  a  Given  Number 

(i)  Let  it  be  required  to  find  from  a  table  the  logarithms  of 
28.6  and  of  326.8. 

The  mantissa  of  the  logarithm  of  28.6  is  found  in  the 
horizontal  line  in  which  28  stands  and  in  the  vertical  column 
at  the  top  of  which  6  stands.     The  mantissa  is  .  4564. 

Since  28.6  lies  between  i  and  100,  the  characteristic 
is  I.  The  entire  logarithm  is  then  i  .4564.  The  meaning  of 
this  is  shown  by  the  explanatory  equation:   10^-4564  =  28.6. 

(2)  To  find  the  logarthim  of  326.8. 

First  find  the  mantissa  of  the  logarithm  of  326  as  before. 
It  is  .5132. 

Since  the  number  326.8  lies  between  326  and  327  the 
mantissa  must  lie  between  the  mantissas  .5132  and  .5145  (the 
mantissa  of  327).  The  interval  between  5132  and  5145  is  13. 
It  is  assumed  that  the  correct  mantissa  of  the  logarithm  of 
326.8  lies  between  5132  and  5145  just  as  326.8  Ues  between 
326  and  327.  But  326 . 8  lies  .8  of  the  way  from  326  up  toward 
327.     But  .8  of  13  equals  10.4,  or  10,  nearly.    Adding  10  to 


Logarithms  175 

5132,  we  have  5142.  The  mantissa  of  the  logarithm  of  326.8 
is,  then,  .5142, 

Since  326.8  lies  between  100  (  =  10*)  and  1,000  (  =  io3), 
the  characteristic  must  be  2,  and  the  entire  logarithm  of 
326. 8is2. 5142.     The  explanatory  equation  is:  10*5142=326. 8. 

Observe  that  to  find  the  characteristic  of  the  logarithm  of 
a  number  it  is  necessary  merely  to  notice  between  which  two 
perfect  powers  of  10  the  given  number  lies. 

(3)  Find  logarithms  of  the  following  numbers:  (i)  13.6; 
(2)  60.5;  (3)  75.22;  (4)  186.2;  (5)  765.3;  (6)  3. 146;  (7) 
1.862;   (8)  2.372;   (9)  12.09. 

§87.     To  Find   from  a  Table  the  Number  that  Corresponds 
to  a  Given  Logarithm 

I.  Let  it  be  required  to  find  the  number  that  corresponds 
to  the  logarithms,  1.6149  and  2.5394. 

Glancing  into  the  body  of  the  table  the  mantissa,  .6149, 
is  found  in  the  horizontal  line  in  which  41  stands  at  the  left 
end  and  2  stands  at  the  top  of  the  vertical  column.  The 
sequence  of  digits  (succession  of  figures)  that  compose  the  re- 
quired number  is,  then,  412. 

Since  the  characteristic  of  the  given  logarithm  (i .  6149) 
is  I,  the  sequence  of  digits  must  he  so  pointed  as  to  make  the 
number  fall  between  10  and  100.  The  number  must  then 
be  41.2.     Explanatory  equation  10^-^*49  =41 .2. 

(2)  To  find  the  number  corresponding  to  the  logarithm, 

2.5394- 

Entering  the  body  of  the  table,  take  out  the  nearest  man- 
tissa to  5394  that  can  be  found,  and  write  down  the  first  three 
digits  as  was  done  above.  The  tabular  mantissa  nearest  to 
5394  is  5391.  The  first  three  digits  corresponding  to  5391, 
in  order,   are  346. 

The  mantissa  next  larger  than  5394  is  5403. 


176  Fir  St -Year  Mathematics 

Arrange  the  numbers  thus: 

Mantissa  of  346=  .5391 

"         "       ?=-5394 

"  347  =-5403 

This  scheme  of  arrangement  shows  the  required  sequence 
of  digits  to  be  between  346  and  347.  Hence,  the  first  three 
digits  sought  must  be  346.  Why?  From  5391  to  5403  is  12, 
and  from  5391  to  5394  is  3.  We  wish  then  to  put  a  sequence 
of  digits  between  346  and  347,  and  j^,  or  \,  of  the  way  from 
346  to  347.     Evidently,  the  sequence  desired  must  be  34625. 

This  sequence  must  be  so  pointed  as  to  give  a  number 
between  10*  (or  100)  and  lo^  (or  1,000),  because  the  given 
logarithm  is  2 .  5394  and  must  correspond  to  a  number  in  the 
interval  from  100  to  1,000. 

The  required  number  is,  then,  346.25. 

3.  Find  numbers  corresponding  to  the  following  logarithms, 
and  write  the  explanatory  equations:  (i)  i . 2672 ;  (2)  i  .7566; 
(3)  2.4942;  (4)3-4871;  (5)3-4876;  (6)2.6520;  (7)1.9507; 
(8)  1.9076;  (9)  2.9673;  (10)  3.9536;  (11)  3-9387;  (12) 
2.8630;   (13)  .8235;   (14)  .8509;   (15)  .8569. 

.         §  88.     Questions  and  Problems 

1.  Show  from  a  four-place  table  the  complete  logarithm 
(characteristic  and  mantissa)  of  47  and  write  the  explanatory 
equation. 

2.  Show  the  complete  logarithms  and  write  the  explana- 
tory equations  for  the  following:  35;  49.5;  54;  59;  68.5; 
73;  86.5;  97.5;  9. 

3.  To  find  the  logarithm  of  the  product  of  67X48. 
Solution:   From  the  table,  log.  7=. 8491,  log.  8=.903i, 

and  log.  56=log.  (7X8)  =  .8451  + .9031=1 .7482. 

Also  from  the  table,  log.  48  =  1.8812,  log.  67  =  1.8261. 


Logarithms  ■I'j^ 

Then  48  =  io*-68" 

and         48X67(=32i6)=io*-'58"Xio*-8a6i=ioi-68i24-i.826i 

=  I03-S073  ;     juSt  aS  lO"  X  lO^  =  IoS  . 

log.  (48X67)=log.  32i6=log. 48+log.  67  =  1.8612  +  1 .8261 

These  examples  illustrate  that  the  logarithm  of  a  product 
equals  the  sum  of  the  logarithms  of  the  factors. 

The  student  may  give  further  examples  from  the  table. 

4.  Find  the  logarithms  of  the  following  products:  (i) 
37X59;  (2)  69X52.5;  (3)  84X88.5;  (4)  17X78.5;  (5) 
45X28X15;   (6)  16X39X48.5- 

5.  To  find  the  logarithm  of  a  quotient  from  the  logarithms 
of  the  numbers  to  be  divided. 

(i)  Find  the  logarithm  of  ^^- ,  or  7  ,  from  log.  63  and  log.  9 
by  use  of  the  table.  From  the  table,  log.  63  =  1.7993,  log. 
9  =.9542,  and  log.  7(=-6/)  =  .645i.  But  i  .7993- .9542  = 
.6451,  or  log.  63 -log.  9=log.  7. 

Again,  from  the  table 

63  =  10^- 7993 

g  =  IO   -9543 

63-f-9  =  IO^-7993-i.io-9542  =  iol-7993  — •9S42  =  io'54Si=7juSt  aS  lO* 
—  I0*  =  I03. 

(2)  Find  the  log.  ,     or  68^19.5. 

From  the  table: 

68        =IO^-833S 

IQ      g=IQl.2900 

68-4-19. 5  =  IO^-832S-MO^-290o  =  ioi-833S— 1.2000  =  10   -5425     just    aS 
IO^-v-IO'»=lo'^—  4  =  10*. 

Or,  log.  (68 -5-19.5)=  log.  68— log.  19.5,  which  illustrates  the 
principle  that  the  logarithm  of  a  quotient  is  the  logarithm  of 
the  dividend  minus  the  logarithm  of  the  divisor. 


178  First-Year  Mathematics 

The  student  may  give  further  illustrations  from  the  table. 

6.  Find  the  logarithms  of  the  following  quotients: 

(i)  53^27;     (2)    48-M3-5;     (3)    87-5^24.5;     (4)    98.5 

^14.5;   (5)  7i-5-^49- 

7.  Find  the  logarithm  of  the  square  of  37.5. 
Solution. — Since  7X7=49  and   log.  7  =.8451  and  log. 

49  =  1 .6902  =  2X  -8451,  we  have  log.  7^  =  2  log.  7.  Show  with 
tables  that  log.  81=2  log.  9. 

Finally,  37.5X37.5  =  io'S74x  10^574=102X1574=  josus. 
The  desired  logarithm  is  3.148,  which  is  twice  the  logarithm 
of  37.5.  This  illustrates  that  the  logarithm  of  any  power 
equals  the  logarithm  of  the  base  times  the  exponent  of  the  power. 

8.  Find  the  logarithm  of  the  cube  37.5. 

Solution:  37. 5X37. 5X37. 5  =  103X1-574  =  104. 722^  just  as 
io»  X 10"  X  10^  =  1(33X2  =  106. 

9.  Find  the  logarithms  of  the  square,  the  cube,  and  the 
fourth  power  of  the  following:  (i)  16.5;  (2)  36.5;  (3)  98; 
(4)  71;  (5)68.5;  and  (6)  99.5. 

10.  Find  the  logarithm  of  the  square  root  of  33. 
Solution:    Let  x  denote  the  desired  logarithm  of  the 

square  root  of  ^2-  Then  io*X  10^=33.  But  33  =  10^518^  and 
io*Xio*=io»*.  Then  io"=*=io''si8,  ^Yhis  will  be  true  if  we 
make  2X=i.5i8  or  :x;=.759,  which  is  the  desired  logarithm. 

Several  such  logarithms  of  selected  tabular  numbers  may 
be  found  by  the  student. 

It  will  be  seen  that  we  may  state  the  following  principle : 

The  logarithm  of  the  square  root  of  any  number  is  one-half 
of  the  logarithm  of  the  number. 

In  a  similar  way  develop  the  principles  for  finding  the 
logarithms  of  the  cube  root  and  of  the  fourth  roots  of  numbers. 

Then  generalize  a  method  for  other  roots  and  test  the 
generalized  method  by  problems. 

The  logarithms  of  some  numbers  may  be  found  from  the 
logarithms  of  other  numbers.    For  example: 


Logarithms  179 

^   =   2X2    =   IO°-3°^°  X   IO°-3°^°  =   io°-30io+o-3oio   =    iQO'Soao 

.*.  log  .  4  =  0.6020. 

6   =   2X3=   IO°-30io  y^   JQO.4771    —   jqO. 3010+0. 4771    —    jQO.7781^ 

.*.  log.  6  =.0.7781 . 

0=2X3=   lO"-*??^   X   lO®-*??!    =   io°-477i+°-477i    =    ioO-954a^ 

.-.  log.  9  =  0.9542. 

14'=  2X7    =   IO°-3°^°   X   lO^-^^Si    =   iqO. 3010+0. 8451   =   jQi.1461^ 

.■.  log  .  14  =  1.1461 . 

10  =  2   X   5    =   IO°-30io   ^   JQO.6990   =   J qO. 3010 +0.6990   =   jqI.oooo 

.•.  log  .  10  =  1 .0000  (as  we  know) . 

126  =  9  X  14  =  IO°-«'S42  X  10^-^461  =  10° •9542  +  1  1461  =  jq2.ioo3^ 

.*.  log  .  126  =  2. .1003  . 

11.  Given  log.  2=0.3010;  log.  3=0.4771;  log.  5=0.6990; 
log.  7=0.8541;  log.  II  =1 .0424;  log.  13  =  1 .1139.  Find  the 
logarithms  of  the  following  numbers: 

(i)  6;  (2)  9;  (3)  4;  (4)  8;  (5)  12;  (6)  14;  (7)  15;  (8)  21; 
(9)  35;  (10)  33;  (11)  42;  (12)  26;  (13)  39;  (14)  55;  (15)  77; 
(16)  65;  (17)  143;  (18)  30;  (19)  70;  (20)  105. 

12.  Find  the  following  quotients  by  the  use  of  logarithms: 

W¥;      (2)H;      (3)  W;      (4)  W- 

13.  Find  the  indicated  roots  of  the  following  by  logarithms: 

(^)  ^^^  '  (4)  Vu4  ;  (6)  V'6^  ; 

(2)V77;  (s)x/T^S;  (y)V^o. 

(3)  1/64  ; 

§  89.     Problems  for  Solution  by  the  Aid  of  Logarithmic  Tables 

Multiplication 

1.  A  lot  is  24.8  by  120.6;  how  many  square  feet  are  there 
in  it  ? 

2.  A  field  is  38.4  rd.  X62.8  rd.;  how  many  square  rods 
in  it? 


i8o  First-Year  Mathematics 

3.  A  street  car  averages  18  trips  a  day,  carrying  36  paid 
passengers  a  trip  at  5  cents  a  fare  for  28  days  a  month.  What 
amount  of  money  is  taken  in  on  this  car  during  the  month. 

Log.  2=0.3010;    log.  3=0.4771  ;    log.  7=0.8451. 

4.  How  many  cubic  feeft  in  a  box  2'. 5X3'  6X4' -2  ? 

5.  How  many  cubic  yards  in  a  room  4.6  yd.XSS  yd.X 
13.3  yd.? 

6.  How  many  cu.  ft.  in  the  room  of  problem  5  ? 

7.  How  many  cu.  ft.  in  a  wagon  box  2' . 9  X3' .  5 X 10' .  2  ? 

8.  Water  weighs  62.5  lb.  per  cu.  ft  and  copper  is  8.8 
times  as  heavy.     How  heavy  is  a  mass  of  copper  2'. 5X3' -87 

X4'-3? 

9.  Ice  weighs  .  92  as  much  as  an  equal  bulk  of  water.  How 
much  does  a  mass  of  ice  6' X  108^X40'  weigh  ? 

10.  What  will  it  cost  to  pave  a  street  .24  yd.  wide  a  dis- 
tance of  I  mile  at  $4 .  78  per  sq.  yd  ?. 

Division 

1.  How  many  sq.  yd.  in  the  lot  of  problem  i  ? 

2.  How  many  acres  in  the  field  of  problem  2  ? 

3.  Find  the  weight  in  ounces  of  a  cu.  in.  of  iron,  a  cu.  ft. 
weighing  487 . 5  pounds. 

4.  How  many  bu.  of  small  grain  will  the  wagon  box  of 
problem  7  contain  if  i  bu.  =  i  .2  cubic  foot? 

5.  How  many  bu.  of  ear  corn  will  the  wagon  box  of  problem 
7  contain  if  i  bu.=2.5  cubic  feet? 

6.  How  many  bu.  of  ear  corn  will  a  crib  1 2' X 16'. 2X40'. 4 
hold  if  2 . 5  cu.  ft.  =  I  bushel  ? 

7.  A  cu.  ft.  of  steel  weighs  490  lb.  Find  the  weight  of  a 
cubic  inch.     How  many  cubic  feet  will  weigh  i  ton  (2,000  lb.)  ? 

Involution 
I .  How  many  cubic  feet  in  a  5-f t.  cube  ?  In  an  8-f t.  cube  ? 
In  a  i6-ft.  cube? 


Logarithms  i8t 

2.  How  many  cubic  feet  are  there  in  a  4.5-ft.  cube?    In 
a  7 . 6-ft.  cube  ?    In  a  24 . 6-foot  cube  ? 

3.  How  many  cubic  inches  are  there  in  a  36".  5  cube?   In 
a  58" .  6  cube  ?    In  a  63" .  7  cube  ? 

Evolution 

1,  Find  the  length  of  the  edge  of  a  cube  whose  volume  is 
216, cu.  ft.;  343  cu.  ft.;  729  cubic  feet. 

2,  Find  the  approximate  length  of  an  edge  of  a  cube  whose 
volume  is  74.09  cu.  ft.;   1092.73  cu.  ft.;  4330.75  cubic  feet. 

3,  Find  by  logarithms  approximately  the  length  of  one  side 
of  the  following  squares: 

(i)  104.04  sq.  ft.;   (2)  357.21  sq.  ft.;    (3)  384.16  sq.  ft.; 
(4)  1004.89  sq.  ft.;  (5)  1497.69  square  feet. 


Un] 


